id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
30,006,616 | OWR-14299911-018 | Comparing Travelling-Salesman and Travelling-Polytope Metrics | For a graph $X$, vertices $x,y$, and a finite vertex set $F$, compare the minimum path length $\operatorname{TS}(x,F,y)$ through every vertex of $F$ with the minimum length $\operatorname{TP}(x,F,y)$ of a path of convex finite-vertex polytopes covering $F$. Determine sharp reverse bounds to $\operatorname{TP}\leq2\oper... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 3 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
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"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-018",
"literature-status:open",
"owr-year:2026"
] | Problem 7 introduces the travelling-salesman and travelling-polytope quantities and asks for their comparison; the report lists only partial bounds and no complete resolution. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | For a graph $X$, vertices $x,y$, and a finite vertex set $F$, compare the minimum path length $\operatorname{TS}(x,F,y)$ through every vertex of $F$ with the minimum length $\operatorname{TP}(x,F,y)$ of a path of convex finite-vertex polytopes covering $F$. Determine sharp reverse bounds to $\operatorname{TP}\leq2\oper... | Problem 7. Travelling polytopes, Genevois Given a graph X, two vertices x, y \in V (X), and a finite set of vertices F \subset V (X), define Median Geometry and Applications 531 Travelling salesman: TS(x, F, y) as the shortest length of a path connect- ing x to y and passing through all the vertices in F ; Travelling pol... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,617 | OWR-14299911-019 | Values of Travelling-Polytope Constants on Median Graphs | Which values can the travelling-polytope constant $\operatorname{TP}(X)$ take as $X$ ranges over median graphs? Characterize the graphs realizing each possible value. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 3 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-019",
"literature-status:open",
"owr-year:2026"
] | The record is only the sentence `A natural question is' and contains no identifiable mathematical claim; it remains open under the queue rule. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Which values can the travelling-polytope constant $\operatorname{TP}(X)$ take as $X$ ranges over median graphs? Characterize the graphs realizing each possible value. | A natural question is: TP(X) := sup\{\alpha | \existsC > 0, \forallx, y, \forallF, TP(x, F, y) + |F | \geq C(TS(x, F, y) + |F |)\alpha \}. Question. Which values can take TP(\cdot) on median graphs? | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,618 | OWR-14299911-020 | Travelling-Polytope Constants for Products of Trees | Compute the travelling-polytope constant $\operatorname{TP}(X)$ when $X$ is a product of regular trees, and more generally determine how this constant behaves under products of median graphs. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 3 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-020",
"literature-status:open",
"owr-year:2026"
] | The extracted record contains the unresolved values-of-TP question and the metric-action question for a wreath product; the source report gives no resolution. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Compute the travelling-polytope constant $\operatorname{TP}(X)$ when $X$ is a product of regular trees, and more generally determine how this constant behaves under products of median graphs. | Question. Which values can take TP(\cdot) on median graphs? For instance, it would be interesting to compute the travelling polytope constant of a product of (regular) trees. Problem 8. Metric actions on median graphs, Genevois An action G y X on a graph is proper if vertex-stabilisers are finite and metrically proper i... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,619 | OWR-14299911-021 | Metric Properness of Wreath-Product Actions on Median Graphs | Let $G$ act properly on a median graph and let $N\trianglelefteq G$. If $G/N$ does not act metrically properly on any median graph, can the permutational wreath product $\mathbb{Z}_2\wr_{G/N}G$ nevertheless act metrically properly on a median graph? Characterize when proper median-graph actions of such wreath products ... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 17 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-021",
"literature-status:open",
"owr-year:2026"
] | Problem 8 asks for the distinction between proper and metrically proper median-graph actions and gives an unresolved wreath-product case; no result settling it was found. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Let $G$ act properly on a median graph and let $N\trianglelefteq G$. If $G/N$ does not act metrically properly on any median graph, can the permutational wreath product $\mathbb{Z}_2\wr_{G/N}G$ nevertheless act metrically properly on a median graph? Characterize when proper median-graph actions of such wreath products ... | Problem 8. Metric actions on median graphs, Genevois An action G y X on a graph is proper if vertex-stabilisers are finite and metrically proper if \{g \in G | d(o, go) \leq R\} is finite for every R \geq 0 (given an arbitrary basepoint o \in V (X)). An interesting problem is to understand the difference between acting pr... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,620 | OWR-14299911-022 | Median Actions and Kazhdan Property for Brin Group 2V | Does Brin's higher-dimensional Thompson group $2V$ admit an action with unbounded orbits on a median graph? Does $2V$ have Kazhdan's property $(T)$? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 17 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-022",
"literature-status:open",
"owr-year:2026"
] | This is the property-(T) part of Problem 9(i) for Brin's group 2V; the report poses it as open and no primary result resolving it was found. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Does Brin's higher-dimensional Thompson group $2V$ admit an action with unbounded orbits on a median graph? Does $2V$ have Kazhdan's property $(T)$? | Does it satisfy Kazhdan’s property (T)? Question ii) Does the Grigorchuk group act metrically properly on a median graph? A construction of Schneeberger shows that a proper action does exist, but this specific action is not metrically proper. Problem 10. Covers of cube complexes, Shepherd The following question has been... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,621 | OWR-14299911-023 | Universal-Cover Isomorphism Algorithms for Cube Complexes | Is there an algorithm that, given two finite nonpositively curved cube complexes, decides whether their universal covers are isomorphic? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-023",
"literature-status:open",
"owr-year:2026"
] | Problem 10(ii) asks for an algorithm deciding isomorphism of universal covers of finite non-positively curved cube complexes; the report notes only the graph case and no general solution was found. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Is there an algorithm that, given two finite nonpositively curved cube complexes, decides whether their universal covers are isomorphic? | Question ii) Is there an algorithm that takes as input a pair of finite non- positively curved cube complexes and determines whether they have isomorphic universal covers? Problem 9. Fixed-point properties, Genevois Inspired by Thompson’s group V as a group of homeomorphisms of the Cantor set C, Brin defined a higher-dim... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,622 | OWR-14299911-024 | Hyperbolic Cubulability, Residual Finiteness, and Local Medianity | If every infinite-order element of a hyperbolic group has unbounded orbits in some cobounded cubical action with quasiconvex hyperplane stabilizers, must the group be cocompactly cubulable or at least a-T-menable? If a hyperbolic group acts properly and cocompactly on a finite-rank median space, must it be residually f... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-024",
"literature-status:open",
"owr-year:2026"
] | The extracted record bundles unresolved cubulability, residual-finiteness, and non-uniform local-median questions from Problems 11 and 12; no complete resolution was found. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
},
{
"url": "https://bhbowditch.com/papers/ch-median.pdf",
"label": "Bowditch preprint"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | If every infinite-order element of a hyperbolic group has unbounded orbits in some cobounded cubical action with quasiconvex hyperplane stabilizers, must the group be cocompactly cubulable or at least a-T-menable? If a hyperbolic group acts properly and cocompactly on a finite-rank median space, must it be residually f... | Is G cocompactly cubulable? Is G necessarily residually finite? Can one remove the word “uniform” from this statement, i.e., instead just suppose that every point has a neighbourhood in which every triple has a unique median? Problem 10. Covers of cube complexes, Shepherd The following question has been posed a while ag... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,623 | OWR-14299911-025 | Cubulability and Residual Finiteness of Hyperbolic Groups | Let $G$ be a hyperbolic group acting properly and cocompactly on a finite-rank median space. Must $G$ be cocompactly cubulable? Must $G$ be residually finite, or even linear? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 17 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-025",
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"owr-year:2026"
] | Problem 11(2) asks whether a hyperbolic group acting properly and cocompactly on a median space is residually finite; the report presents this as open and no resolution was found. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Let $G$ be a hyperbolic group acting properly and cocompactly on a finite-rank median space. Must $G$ be cocompactly cubulable? Must $G$ be residually finite, or even linear? | Is G cocompactly cubulable? Is G necessarily residually finite? Problem 12. Cartan-Hadamard theorem for median spaces, Bowditch One can show that any complete simply connected path-metric space which is uniformly locally median is in fact median. Here, “uniformly locally median” means that there is some ǫ > 0 such that ... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,624 | OWR-14299911-026 | Cartan-Hadamard Theorem without Uniform Local Medianity | Let $X$ be a complete simply connected path-metric space in which every point has a neighborhood where each triple has a unique median. Must $X$ be a median space even when the radii of these neighborhoods are not uniformly bounded below? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
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"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-026",
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] | Bowditch's Cartan-Hadamard question asks whether pointwise local uniqueness of medians suffices without a uniform radius; the cited preprint and report leave it unresolved. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
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"url": "https://bhbowditch.com/papers/ch-median.pdf",
"label": "Bowditch preprint"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Let $X$ be a complete simply connected path-metric space in which every point has a neighborhood where each triple has a unique median. Must $X$ be a median space even when the radii of these neighborhoods are not uniformly bounded below? | Can one remove the word “uniform” from this statement, i.e., instead just suppose that every point has a neighbourhood in which every triple has a unique median? (1) Suppose that for all infinite-order g \in G, there exists a cobounded (not necessarily proper) action of G on a CAT(0) cube complex Xg such that the hyperp... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,625 | OWR-14299911-027 | Asymptotic-Cone Characterization of Hierarchically Hyperbolic Spaces | Characterize, through their asymptotic cones, the coarse median groups that are quasi-isometric to hierarchically hyperbolic spaces. Specifically, is a coarse median group quasi-isometric to a hierarchically hyperbolic space if and only if each nonprincipal asymptotic cone is bi-Lipschitz equivalent to a universal real... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 3 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
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"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-027",
"literature-status:open",
"owr-year:2026"
] | Problem 13(ii) proposes an asymptotic-cone characterization of coarse median groups quasi-isometric to HHSs; the report states it as a question and no resolution was found. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Characterize, through their asymptotic cones, the coarse median groups that are quasi-isometric to hierarchically hyperbolic spaces. Specifically, is a coarse median group quasi-isometric to a hierarchically hyperbolic space if and only if each nonprincipal asymptotic cone is bi-Lipschitz equivalent to a universal real... | Can we characterise groups quasi-isometric to an HHS within the class of coarse median groups in terms of their asymptotic cones? Problem 14. Graphical apiculate algebras, Chepoi \& Bandelt Every connected graph G = (V, E) can be turned into a ternary algebra, called an apex algebra of G: an apex operation (. . .) : V ... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,626 | OWR-14299911-028 | Characterization of Graphical Apiculate and Generalized Median Algebras | A ternary algebra is called apiculate if it is the apex algebra of an apiculate graph, and graphical if it and all of its subalgebras are apiculate algebras. Characterize the apiculate graphs whose apex algebras are graphical. Likewise, characterize the generalized median graphs whose generalized median algebras are gr... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 4 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
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"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-028",
"literature-status:open",
"owr-year:2026"
] | The record is only the setup of Problem 14 on graphical apiculate algebras and contains no complete claim to verify; the associated characterization questions remain open. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | A ternary algebra is called apiculate if it is the apex algebra of an apiculate graph, and graphical if it and all of its subalgebras are apiculate algebras. Characterize the apiculate graphs whose apex algebras are graphical. Likewise, characterize the generalized median graphs whose generalized median algebras are gr... | Problem 14. Graphical apiculate algebras, Chepoi \& Bandelt Every connected graph G = (V, E) can be turned into a ternary algebra, called an apex algebra of G: an apex operation (. . .) : V 3 \to V maps any triplets u, v, w and Problem 13. Universal real cubings, Casals-Ruiz (from discussions with M. Hagen and I. Kazac... | corrected_verified | The official Oberwolfach report was checked. The mathematical question was isolated from surrounding exposition and its notation was repaired for self-contained publication. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,627 | OWR-14299911-029 | Corner Peeling in Cubical Subcomplexes of Integer Three-Space | Let $G$ be a finite connected subgraph of $\mathbb{Z}^3$, and let $X(G)$ be the cubical complex containing exactly those squares and three-cubes whose vertices and edges all belong to $G$. Assume that $X(G)$ is simply connected and that every connected component of the intersection of $X(G)$ with a coordinate hyperplan... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 6 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | {
"source_url": "https://zenodo.org/records/23049729",
"checked_at": "2026-09-30",
"kind": "claim",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 unrefereed claim, 2026-09-30] New unrefereed preprint addressing OWR-14299911-029: (https://zenodo.org/records/23049729) | [
"owr",
"oberwolfach-reports",
"OWR-14299911-029",
"literature-status:open",
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] | The corner-peeling assertion for simply connected subcomplexes of \mathbin{\mathbb{Z}^3} with simply connected pieces is stated as a conjecture; the cited 2026 work gives only an additional-condition theorem, not a proof of the conjecture. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
"label": "OWR report"
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{
"url": "https://arxiv.org/abs/2602.12894",
"label": "related 2026 preprint"
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | Let $G$ be a finite connected subgraph of $\mathbb{Z}^3$, and let $X(G)$ be the cubical complex containing exactly those squares and three-cubes whose vertices and edges all belong to $G$. Assume that $X(G)$ is simply connected and that every connected component of the intersection of $X(G)$ with a coordinate hyperplan... | Problem 15. Corners in subcomplexes of Z3 , V. Chepoi (from discussions with J. Chalopin and M. Kokkou) Let G be a finite connected (partial) subgraph of Z3 . Let X(G) be the cube complex of G, i.e., a square (respectively, a 3-cube) of Z3 belongs to X(G) if and only if all 4 vertices and 4 edges of this square belong t... | corrected_verified | The official Oberwolfach report was checked. The mathematical question was isolated from surrounding exposition and its notation was repaired for self-contained publication. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,628 | OWR-14299911-031 | Property T for Random Free-Product Quotients above One-Third Density | For a free product of at least three nontrivial factors, does a random quotient in the free-product density model have Kazhdan's property $(T)$ with high probability at every density $d>1/3$? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540](https://doi.org/10.4171/owr/2026/8)
## Literature review (checked 2026-08-22)
**Status:... | 3 | open | null | 2,026 | 19 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
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"description": "Problems involving probability theory, stochastic processes, and random structures.",
"slug": "probability",
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"created_at": "2026-07-31T15:26:25.671Z"
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"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299911-031",
"literature-status:open",
"owr-year:2026"
] | The free-product-density Property (T) question for at least three factors is explicitly left open in the 2026 report and in the 2025 random-quotients paper; no resolution was found. | [
{
"url": "https://ems.press/content/serial-article-files/53603",
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{
"url": "https://arxiv.org/abs/2502.08630",
"label": "Einstein--Krishna--Montee--Ng--Steenbock preprint"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/8 | null | null | null | For a free product of at least three nontrivial factors, does a random quotient in the free-product density model have Kazhdan's property $(T)$ with high probability at every density $d>1/3$? | Does this remain true for a random quotient of a free product of at least 3 factors in the free product density model at density d > 1/3? The report passage is retained verbatim as the relevant context for this extracted problem. | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540 |
30,006,629 | OWR-14299912-001 | Nonlinear-Waves Workshop Introduction Fragment | conjecture. (2) Probabilistic methods. For example, Björn Bringmann presented recent re- sults on the the global well-posedness of singular stochastic PDEs of para- bolic type and explained how the methods connect to dispersive problems. In a different direction, Nikolay Tzvetkov’s lecture on probabilistic well- posedn... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Nonlinear Waves and Dispersive Equations (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/10, pp. 611--682](https://doi.org/10.4171/owr/2026/10)
## Literature review (checked 2026-08-22)
... | 3 | open | null | 2,026 | 9 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
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"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
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"created_at": "2026-07-31T15:26:25.671Z"
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"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | Unrecoverable OWR statement (The official report confirms that this extraction is a truncated workshop-summary fragment containing no identifiable conjecture. No separat) | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299912-001",
"literature-status:open",
"owr-year:2026"
] | The record is only a truncated occurrence of the word `conjecture' in workshop prose and identifies no mathematical claim; it remains open under the queue rule. | [
{
"url": "https://publications.mfo.de/bitstream/handle/mfo/4444/OWR_2026_10.pdf?sequence=1",
"label": "OWR report"
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/10 | null | null | null | null | conjecture. (2) Probabilistic methods. For example, Björn Bringmann presented recent re- sults on the the global well-posedness of singular stochastic PDEs of para- bolic type and explained how the methods connect to dispersive problems. In a different direction, Nikolay Tzvetkov’s lecture on probabilistic well- posedn... | unrecoverable | The official report confirms that this extraction is a truncated workshop-summary fragment containing no identifiable conjecture. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. | null | null | null | null | null | Nonlinear Waves and Dispersive Equations (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/10, pp. 611--682 |
30,006,630 | OWR-14299912-002 | Blowup Residues, Stability, and Rates for Mass-Critical gKdV | For the mass-critical generalized Korteweg-de Vries equation, determine whether every finite-time, finite-point blowup solution with rate exponent $\nu\in(1/3,1/2)$ has an $H^1$ blowup residue, and whether the finite-time, finite-point blowup family constructed for $\nu\in(3/7,1/2)$ is stable. Determine whether the onl... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Nonlinear Waves and Dispersive Equations (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/10, pp. 611--682](https://doi.org/10.4171/owr/2026/10)
## Literature review (checked 2026-08-22)
... | 3 | open | null | 2,026 | 9 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
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"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299912-002",
"literature-status:open",
"owr-year:2026"
] | The mass-critical gKdV report explicitly says both the existence of the requested H^{1} blowup residue and stability of the constructed solutions are open, even for the known rate \nu=2/5. | [
{
"url": "https://publications.mfo.de/bitstream/handle/mfo/4444/OWR_2026_10.pdf?sequence=1",
"label": "OWR report"
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{
"url": "https://arxiv.org/abs/2601.20801",
"label": "Martel--Pilod preprint"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/10 | null | null | null | For the mass-critical generalized Korteweg-de Vries equation, determine whether every finite-time, finite-point blowup solution with rate exponent $\nu\in(1/3,1/2)$ has an $H^1$ blowup residue, and whether the finite-time, finite-point blowup family constructed for $\nu\in(3/7,1/2)$ is stable. Determine whether the onl... | Below, we list some interesting open questions regarding the blowup phenome- non for the mass critical gKdV equation. (i) Does there exist a blowup residue r\alpha in H 1 for any solution of (1) blowing up in finite time at a finite point (thus, ν \in ( 13 , 21 ))? This question is open even for the solution constructed ... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Nonlinear Waves and Dispersive Equations (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/10, pp. 611--682 |
30,006,631 | OWR-14299912-003 | Higher Sobolev Norm Growth for Half-Wave Maps | For nonrational initial data for the half-wave maps equation on the circle, determine the behavior of the higher Sobolev norms $H^s$ for $s>1/2$. Can an $H^s$ norm blow up in finite or infinite time? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Nonlinear Waves and Dispersive Equations (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/10, pp. 611--682](https://doi.org/10.4171/owr/2026/10)
## Literature review (checked 2026-08-22)
... | 3 | open | null | 2,026 | 9 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
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"id": 3,
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"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299912-003",
"literature-status:open",
"owr-year:2026"
] | For half-wave maps, the cited work proves global well-posedness at critical H^{1/2}, while the behavior of higher H^{s} norms for non-rational data is explicitly left open. | [
{
"url": "https://publications.mfo.de/bitstream/handle/mfo/4444/OWR_2026_10.pdf?sequence=1",
"label": "OWR report"
},
{
"url": "https://arxiv.org/abs/2602.20985",
"label": "Gérard--Lenzmann preprint"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/10 | null | null | null | For nonrational initial data for the half-wave maps equation on the circle, determine the behavior of the higher Sobolev norms $H^s$ for $s>1/2$. Can an $H^s$ norm blow up in finite or infinite time? | Note that Gr1 (C2 ) \sim = S2. • In [5] we prove global well-posedness for (HWM) posed on the real line R and all initial data in Ḣ 1/2 (R; S 2 ) along with the soliton resolution. • It is an interesting open question to understand the behavior of higher Sobolev norms H s with s > 1/2 for non-rational data. Is blowup ... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Nonlinear Waves and Dispersive Equations (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/10, pp. 611--682 |
30,006,632 | OWR-14299913-001 | Orthogonal Symmetry Conjecture for Rigid Higher Categories | Let $\operatorname{Cat}^{\mathrm{rigid}}_{(\infty,n)}(\mathcal{V})$ be the $\infty$-category of rigid $(\infty,n)$-categories enriched over a presentably symmetric monoidal $\infty$-category $\mathcal{V}$. Taking opposites in each categorical tier gives an action of $O(1)^n$. Prove that this action extends to an action... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | open | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-001",
"literature-status:open",
"owr-year:2026"
] | The extracted record names an unidentified conjecture; the originating report describes the relevant orthogonal-group extension as conjectural and ongoing, with no proof reported. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2205.04954",
"label": "Ayala--Francis, Symmetries of a rigid braided category"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Let $\operatorname{Cat}^{\mathrm{rigid}}_{(\infty,n)}(\mathcal{V})$ be the $\infty$-category of rigid $(\infty,n)$-categories enriched over a presentably symmetric monoidal $\infty$-category $\mathcal{V}$. Taking opposites in each categorical tier gives an action of $O(1)^n$. Prove that this action extends to an action... | This conjecture is logically related to, but not implied by, the cobordism hypoth- esis, as I will indicate. Generally, this conjecture is expected to be true based on heuristics of topological defects/operators in a quantum field theory on Rn . Orthogonal symmetries of higher categories David Ayala (joint work with Joh... | corrected_verified | The official Oberwolfach report was checked. The mathematical question was isolated from surrounding exposition and its notation was repaired for self-contained publication. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,633 | OWR-14299913-002 | Higher-Categorical Models of Continuous and Spacetime Symmetries | Develop precise higher-categorical data and constructions that recover standard continuous internal symmetries and spacetime symmetries of quantum field theories, and establish their functorial and physical properties. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | open | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
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"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-002",
"literature-status:open",
"owr-year:2026"
] | The report explicitly says that precise reformulations of standard continuous and spacetime symmetries remain largely open; later work on universal non-invertible symmetries likewise leaves the general higher-categorical definition unsettled. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://doi.org/10.1002/prop.202200143",
"label": "Bhardwaj et al., Universal Non-Invertible Symmetries"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Develop precise higher-categorical data and constructions that recover standard continuous internal symmetries and spacetime symmetries of quantum field theories, and establish their functorial and physical properties. | However, finding a precise technique for reformulating various standard examples of symmetries (continuous symmetries, spacetime symmetries) in terms of such data has largely remained an open question. Symmetry Theories from the Universal Bulk Ilka Brunner (joint work with Severin König, Ingmar Saberi) Higher Structure... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,634 | OWR-14299913-003 | Two-Categorical Interpretation of a Quantum Lattice Model | Give a categorical interpretation of the computed symmetry structure in the two-dimensional quantum lattice model, and determine whether it is naturally governed by a two-category. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | open | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-003",
"literature-status:open",
"owr-year:2026"
] | The report leaves the categorical interpretation of the directly computed two-dimensional lattice example open; recent sector-category constructions address particular models but do not supply the requested general 2-category connection. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://doi.org/10.1007/s00220-025-05363-w",
"label": "Bols et al., Classification of the Anyon Sectors of Kitaev's Quantum Double Model"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Give a categorical interpretation of the computed symmetry structure in the two-dimensional quantum lattice model, and determine whether it is naturally governed by a two-category. | By direct calculation we also found an example in a 2d quantum lattice model, but here the connection with categories remains an open problem. Could a 2-category be involved? Higher Structures from Symmetries in Quantum Field Theory 755 | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,635 | OWR-14299913-004 | Liouville Theory from the Virasoro Conformal Net | Prove that applying the fully extended two-dimensional conformal-field-theory construction to the Virasoro conformal net produces Liouville conformal field theory. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | open | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-004",
"literature-status:open",
"owr-year:2026"
] | The Virasoro-net construction is still presented as conjecturally producing Liouville theory; recent work proves a net-to-unitary-VOA direction but not the claimed identification with Liouville theory. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2507.20735",
"label": "Henriques--Tener, Every conformal net has an associated unitary VOA"
},
{
"url": "https://arxiv.org/abs/2204.02745",
"label": "Vargas, T... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Prove that applying the fully extended two-dimensional conformal-field-theory construction to the Virasoro conformal net produces Liouville conformal field theory. | We conjecture that, when applied to the Virasoro conformal net, our construc- tion produces Liouville theory. Construction of fully extended 2d CFTs via conformal net methods André Henriques (joint work with James Tener) | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,636 | OWR-14299913-005 | Extending Trisection Invariants to Four-Dimensional TQFTs | Extend the trisection invariants of smooth four-manifolds to a four-dimensional topological quantum field theory, or identify an obstruction to such an extension. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | open | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-005",
"literature-status:open",
"owr-year:2026"
] | The originating authors state that extending their trisection invariant to a four-dimensional TQFT is unknown; subsequent trisection-invariant work does not establish such an extended TQFT. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2511.19384",
"label": "Meusburger--Mulevicius--Torzewska, Categorical 4-manifold invariants from trisection diagrams"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Extend the trisection invariants of smooth four-manifolds to a four-dimensional topological quantum field theory, or identify an obstruction to such an extension. | Open Questions • It is unknown to us, if our trisection invariants of 4-manifolds can be extended to a 4d TQFT. The report passage is retained verbatim as the relevant context for this extracted problem. | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,637 | OWR-14299913-006 | Finite-Depth Channel Classification of Mixed-State Topological Order | Classify mixed-state topological orders under finite-depth quantum channels: given two states, determine whether a finite-depth channel maps one to the other, and characterize the obstructions supplied by anyons and nontrivial braiding. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | partially_solved | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-006",
"literature-status:partially-solved",
"owr-year:2026"
] | Recent work gives a partial classification of mixed-state topological order under quasilocal quantum channels and proves obstruction results for nontrivial braiding, but a complete finite-depth-channel criterion remains conjectural. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://doi.org/10.1103/PRXQuantum.6.010315",
"label": "Lee et al., Toward a Classification of Mixed-State Topological Orders"
},
{
"url": "https://arxiv.org/abs/2501.02398",
"label... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Classify mixed-state topological orders under finite-depth quantum channels: given two states, determine whether a finite-depth channel maps one to the other, and characterize the obstructions supplied by anyons and nontrivial braiding. | These anyons, if the braiding is non-trivial, can work as an index in the classification of mixed state topological order, where we ask if there is a finite-depth quantum channel to map one state to the other[2]. Topological Order: The Operator-Algebraic Approach Yoshiko Ogata | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,638 | OWR-14299913-008 | Higher-Dimensional Quantum Cellular Automata beyond Translations | Classify nontrivial elements in the quotient of higher-dimensional quantum cellular automata by finite-depth circuits and shifts. In particular, determine which elements occur beyond lattice translations and the currently known three-dimensional examples. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | partially_solved | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-008",
"literature-status:partially-solved",
"owr-year:2026"
] | Explicit higher-dimensional QCAs beyond lattice translation are now constructed, including dimension-periodic Clifford and non-Clifford families; a complete classification of all interesting quotient elements is not established. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://doi.org/10.1103/4519-v15s",
"label": "Sun et al., Clifford quantum cellular automata from TQFTs"
},
{
"url": "https://arxiv.org/abs/2607.21697",
"label": "Sun et al., Non-Cl... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Classify nontrivial elements in the quotient of higher-dimensional quantum cellular automata by finite-depth circuits and shifts. In particular, determine which elements occur beyond lattice translations and the currently known three-dimensional examples. | A natural question is whether there are anything else interesting in the quo- tient other than lattice translation in higher dimensions. [3] constructed the first example of a QCA in 3 spatial dimensions. Quantum Cellular Automata from TQFT Nathanan Tantivasadakarn (joint work with Meng Sun, Bowen Yang, Zongyuan Wang an... | corrected_verified | Only clear formula delimiters, notation, grammar, and typography were repaired; the mathematical content and scope are preserved. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,639 | OWR-14299913-009 | Quantum Cellular Automata Realizing the Full Witt Group | Construct higher-dimensional quantum cellular automata corresponding to every element of the Witt group of nondegenerate braided fusion categories, extending the existing constructions for its pointed part. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794](https://doi.org/10.4171/owr/2026/12)
## Literature review (che... | 3 | open | null | 2,026 | 16 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299913-009",
"literature-status:open",
"owr-year:2026"
] | The non-constant-depth nature of the constructed QCAs and the realization of the full Witt group remain unresolved in general; existing papers provide constructions and conditional or partial classification results only. | [
{
"url": "https://doi.org/10.4171/owr/2026/12",
"label": "originating Oberwolfach report"
},
{
"url": "https://doi.org/10.1103/PRXQuantum.3.030326",
"label": "Shirley et al., Three-dimensional quantum cellular automata"
},
{
"url": "https://doi.org/10.1103/4519-v15s",
"label": "Sun e... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/12 | null | null | null | Construct higher-dimensional quantum cellular automata corresponding to every element of the Witt group of nondegenerate braided fusion categories, extending the existing constructions for its pointed part. | A remaining open question is to show that these QCAs cannot be realized as constant depth circuits. More generally, it remains open whether there are QCAs corresponding to the Witt group of non-degenerated braided fusion categories [9], to which what was covered here is only the pointed part. The report passage is reta... | corrected_verified | The official Oberwolfach report was checked. The mathematical question was isolated from surrounding exposition and its notation was repaired for self-contained publication. | null | null | null | null | null | Higher Structures from Symmetries in Quantum Field Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/12, pp. 737--794 |
30,006,640 | OWR-14299914-001 | Geometry and Sampling on Probability Manifolds | Compute geometric quantities, including Riemannian and sectional curvatures, for general thermodynamical probability manifolds arising from master equations, develop the associated network analysis of curvature tensors, and use these calculations to design sampling algorithms for distributions on simplices. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884](https://doi.org/10.4171/owr/2026/13)
## Literature review (... | 3 | partially_solved | null | 2,026 | 6 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299914-001",
"literature-status:partially-solved",
"owr-year:2026"
] | Riemannian, sectional, Ricci, and scalar curvature calculations on several probability manifolds are available, but a general algorithmic theory for sampling on simplex-supported distributions is not established. | [
{
"url": "https://doi.org/10.4171/owr/2026/13",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2504.19368",
"label": "Li, Geometric calculations on probability manifolds"
},
{
"url": "https://arxiv.org/abs/1803.06360",
"label": "Li, Transport information... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/13 | null | null | null | Compute geometric quantities, including Riemannian and sectional curvatures, for general thermodynamical probability manifolds arising from master equations, develop the associated network analysis of curvature tensors, and use these calculations to design sampling algorithms for distributions on simplices. | What are geometric quantities, including Riemannian and sectional curvatures, in probability manifolds? Can we use these computations to design algorithms in sampling distributions supported on simplex sets? Geometric calculations on probability manifolds from reciprocal relations in Master equations Wuchen Li | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884 |
30,006,641 | OWR-14299914-002 | Causal-Attention Retrieval Motivation | For a distinguished site i∗ = ⌊\sigma0 N ⌋, with \sigma0 \in (0, 1), we ask whether the last token \thetaN (t) at time t decodes to the initial token at position i∗ . Large language models based on causal transformers are routinely deployed in regimes where the available context is much longer than the part of the prom... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884](https://doi.org/10.4171/owr/2026/13)
## Literature review (... | 3 | partially_solved | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | Unrecoverable OWR statement (The official report uses this question to define its retrieval observable and then derives an asymptotic formula and a rigorous U-shaped ret) | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299914-002",
"literature-status:partially-solved",
"owr-year:2026"
] | The causal-attention model now has rigorous mean-field and next-order correlation results and a U-shaped retrieval theorem under explicit assumptions, but this does not settle exact decoding for the full transformer setting or all parameters. | [
{
"url": "https://doi.org/10.4171/owr/2026/13",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2605.09213",
"label": "Duerinckx--Geshkovski--Rossi, Kinetic theory for Transformers"
},
{
"url": "https://doi.org/10.1162/tacl_a_00638",
"label": "Liu et al.,... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/13 | null | null | null | null | For a distinguished site i∗ = ⌊\sigma0 N ⌋, with \sigma0 \in (0, 1), we ask whether the last token \thetaN (t) at time t decodes to the initial token at position i∗ . Large language models based on causal transformers are routinely deployed in regimes where the available context is much longer than the part of the prom... | unrecoverable | The official report uses this question to define its retrieval observable and then derives an asymptotic formula and a rigorous U-shaped retrieval theorem for the stated model; it is not left as a standalone open problem. No separate clean statement is promoted; the original record remains publicly visible and retained... | null | null | null | null | null | Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884 |
30,006,642 | OWR-14299914-003 | Metastability and Massive-Arratia Limits for Weakly Interacting Diffusions | Prove dynamical metastability for the mean-field partial differential equation governing weakly interacting diffusions. Also prove convergence of the particle system to the massive Arratia flow in the joint regime $N\to\infty$, $\gamma\to\infty$, $\ell\to 0$, and $\gamma\ell/\log N\to\infty$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884](https://doi.org/10.4171/owr/2026/13)
## Literature review (... | 3 | open | null | 2,026 | 19 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 19,
"name": "probability",
"display_name": "Probability",
"description": "Problems involving probability theory, stochastic processes, and random structures.",
"slug": "probability",
"order_index": 19,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299914-003",
"literature-status:open",
"owr-year:2026"
] | The originating work explicitly identifies rigorous convergence to the massive Arratia flow in the stated joint scaling regime as an open question; existing Arratia-flow papers study the limit model rather than this particle-system justification. | [
{
"url": "https://doi.org/10.4171/owr/2026/13",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/1504.00559",
"label": "Konarovskyi--von Renesse, Modified massive Arratia flow"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/13 | null | null | null | Prove dynamical metastability for the mean-field partial differential equation governing weakly interacting diffusions. Also prove convergence of the particle system to the massive Arratia flow in the joint regime $N\to\infty$, $\gamma\to\infty$, $\ell\to 0$, and $\gamma\ell/\log N\to\infty$. | Furthermore, the rigor- ous justification of the limit to the massive Arratia flow [6] in the scaling regime N \to \infty, \gamma \to \infty, ℓ \to 0 : \gammaℓ/ log N \to \infty is a challenging open question. (E2) Coalescence of clusters (tcoalesce \sim N ). After initial clustering, K \approx kmax (k) clusters have for... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884 |
30,006,643 | OWR-14299914-004 | Sharper Decay and Algorithms for Non-Gaussian Targets | For the projected gradient dynamics with a non-Gaussian log-concave target, prove a sharper decay estimate that does not deteriorate with the convexity parameter $\lambda$, and construct a provably convergent numerical scheme. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884](https://doi.org/10.4171/owr/2026/13)
## Literature review (... | 3 | open | null | 2,026 | 19 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 19,
"name": "probability",
"display_name": "Probability",
"description": "Problems involving probability theory, stochastic processes, and random structures.",
"slug": "probability",
"order_index": 19,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299914-004",
"literature-status:open",
"owr-year:2026"
] | The report leaves both a sharper non-Gaussian decay estimate and a good provably convergent numerical scheme open; available algorithms for related Wasserstein problems do not prove these claims for the stated model. | [
{
"url": "https://doi.org/10.4171/owr/2026/13",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2505.24384",
"label": "Provably convergent stochastic fixed-point algorithm for Wasserstein barycenters"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/13 | null | null | null | For the projected gradient dynamics with a non-Gaussian log-concave target, prove a sharper decay estimate that does not deteriorate with the convexity parameter $\lambda$, and construct a provably convergent numerical scheme. | We conjecture that the convergence result above is far from optimal, since, unlike when \pi is Gaussian, it depends on the parameter \lambda > 0. A better decay estimate remains open, as does a good (provably convergent) numerical scheme for non-Gaussian targets. and one still has a projected gradient structure on G. T... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884 |
30,006,644 | OWR-14299914-005 | Convergence of Accelerated Sampling on Probability Spaces | Prove convergence of the proposed accelerated fixed-point and Anderson-mixing methods on suitable regular subsets of general probability measures, including the Stein-geometry sampling algorithm for smooth densities. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884](https://doi.org/10.4171/owr/2026/13)
## Literature review (... | 3 | partially_solved | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299914-005",
"literature-status:partially-solved",
"owr-year:2026"
] | Convergence theory has been proved for the Gaussian Bures--Wasserstein specialization, while the proposed regular-subset method and convergence theory on general probability-measure spaces remain open. | [
{
"url": "https://doi.org/10.4171/owr/2026/13",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2601.22038",
"label": "Aksenov--Eigel--Oster, Anderson Mixing in Bures--Wasserstein Space"
},
{
"url": "https://doi.org/10.1007/s10915-025-03027-4",
"label": "... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/13 | null | null | null | Prove convergence of the proposed accelerated fixed-point and Anderson-mixing methods on suitable regular subsets of general probability measures, including the Stein-geometry sampling algorithm for smooth densities. | To sidestep this problem, we propose to consider a more regular subset of measures or tangent vectors. The theoretical proof of convergence, however, remains an open question. Accelerated Fixed-point Iteration over Spaces of Probability Measures Vitalii Aksenov (joint work with M. Eigel and M. Oster) | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Flows on Measure Spaces and Applications in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/13, pp. 795--884 |
30,006,645 | OWR-14299915-001 | Cohomological Finite-Generation Survey Summary | The talk focused on recent results and stating open questions related to the finite generation of the co- homology of finite group schemes over Verlinde categories. Cohomology of Finite Groups: Interactions and Applications 543 | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 4 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | Unrecoverable OWR statement (The official report confirms that this extraction is a programmatic summary of a survey and does not state a standalone open problem. No sep) | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-001",
"literature-status:open",
"owr-year:2026"
] | The record is a programmatic description of recent results and open questions, not a resolved finite-generation theorem for finite group schemes over Verlinde categories; the general problem remains active. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2510.02908",
"label": "Notes on Cohomological Finite Generation for Finite Group Schemes"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | null | The talk focused on recent results and stating open questions related to the finite generation of the co- homology of finite group schemes over Verlinde categories. Cohomology of Finite Groups: Interactions and Applications 543 | unrecoverable | The official report confirms that this extraction is a programmatic summary of a survey and does not state a standalone open problem. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,646 | OWR-14299915-002 | Etingof-Ostrik Cohomological Finite-Generation Conjecture | Let $\mathcal C$ be a finite tensor category. Prove that $H^*(\mathcal C)=\operatorname{Ext}^*_{\mathcal C}(\mathbf 1,\mathbf 1)$ is Noetherian and that $\operatorname{Ext}^*_{\mathcal C}(M,N)$ is a finite $H^*(\mathcal C)$-module for all objects $M,N\in\mathcal C$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 4 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-002",
"literature-status:partially-solved",
"owr-year:2026"
] | The Etingof--Ostrik CFG conjecture remains open for arbitrary finite tensor categories, although it is proved in important classes including braided categories and several Hopf-algebraic cases. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://doi.org/10.1112/blms.13048",
"label": "Bergh, Support varieties without the tensor product property"
},
{
"url": "https://doi.org/10.1007/s00209-023-03309-3",
"label": "Separ... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | Let $\mathcal C$ be a finite tensor category. Prove that $H^*(\mathcal C)=\operatorname{Ext}^*_{\mathcal C}(\mathbf 1,\mathbf 1)$ is Noetherian and that $\operatorname{Ext}^*_{\mathcal C}(M,N)$ is a finite $H^*(\mathcal C)$-module for all objects $M,N\in\mathcal C$. | Conjecture 1.2 (Etingof-Ostrik’04). Let C be a finite tensor category. (1) H ∗ (C) = Ext∗C (1, 1) is Noetherian; (2) For any M, N \in C, Ext∗C (M, N ) is a finite module over Ext∗C (1, 1). When conjecture holds, we say that C satisfies CFG (Cohomological Finite Generation). This is an essential property to be able to stud... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,647 | OWR-14299915-003 | Balmer Spectra of Stable Finite Tensor Categories | For every finite tensor category $\mathcal C$, identify the Balmer spectrum of $\operatorname{stab}\mathcal C$ with the projective spectrum of the categorical center $Z_{\mathcal C}(H^*(\mathcal C))$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 4 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-003",
"literature-status:partially-solved",
"owr-year:2026"
] | Comparison maps to the categorical center and partial support/spectrum results are established under finite-generation hypotheses and in several classes, but the stated identification for every finite tensor category remains conjectural. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2112.11170",
"label": "Nakano--Vashaw--Yakimov, On the spectrum and support theory"
},
{
"url": "https://arxiv.org/abs/2506.19946",
"label": "Nakano--Vashaw--Ya... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | For every finite tensor category $\mathcal C$, identify the Balmer spectrum of $\operatorname{stab}\mathcal C$ with the projective spectrum of the categorical center $Z_{\mathcal C}(H^*(\mathcal C))$. | Conjecture 1.3 (Nakano-Vashaw-Yakimov, 2021). Let C be a finite tensor cate- gory. Then the Balmer spectrum of the stable category of C can be identified with a certain subring Zc (H ∗ (C)) of H ∗ (C), called the “categorical center”. When conjecture holds, we say that C satisfies CFG (Cohomological Finite Generation). Th... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,648 | OWR-14299915-004 | Revised Donkin Tilting-Module Conjectures | Let $G$ be a simple simply connected algebraic group over an algebraically closed field of characteristic $p>0$, with root system $\Phi$ and Coxeter number $h$. Prove that Donkin’s Tilting Module Conjecture holds for every prime $p$ exactly when $\Phi$ has type $A_n$ or $B_2$, and prove that it holds whenever $p>h$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 4 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-004",
"literature-status:open",
"owr-year:2026"
] | The extracted statement is a broad lifting problem rather than a theorem; Donkin-related lifting conjectures have substantial case-by-case progress and counterexamples to nearby formulations, but no general resolution. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2103.14164",
"label": "Bendel--Nakano--Pillen--Sobaje, Donkin's Tilting Module Conjecture I"
},
{
"url": "https://arxiv.org/abs/2107.11615",
"label": "Bendel--N... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | Let $G$ be a simple simply connected algebraic group over an algebraically closed field of characteristic $p>0$, with root system $\Phi$ and Coxeter number $h$. Prove that Donkin’s Tilting Module Conjecture holds for every prime $p$ exactly when $\Phi$ has type $A_n$ or $B_2$, and prove that it holds whenever $p>h$. | In the study of modular representations of Lie algebras arising from algebraic groups, a fundamental problem is determining when such representations lift to the ambient algebraic group. Cohomology for Algebraic Groups and Donkin’s Conjectures Daniel K. Nakano (joint work with Christopher P. Bendel, Cornelius Pillen, P... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,649 | OWR-14299915-005 | Abstract Cohomology of Pro-$p$ Groups in Cross Characteristic | Let $G$ be a pro-$p$ group and let $q\neq p$ be a prime. Must the abstract-group cohomology $H^n_{\mathrm{abs}}(G;\mathbb F_q)$ vanish for every $n>0$, as continuous cohomology does? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 17 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-005",
"literature-status:open",
"owr-year:2026"
] | The question concerns abstract cohomology of pro-p groups with coefficients of characteristic different from p; the report records it as unresolved, and the cited discrete-cohomology work does not settle the general vanishing claim. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/1705.09131",
"label": "Ivanov--Mikhailov, On discrete homology of a free pro-p-group"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | Let $G$ be a pro-$p$ group and let $q\neq p$ be a prime. Must the abstract-group cohomology $H^n_{\mathrm{abs}}(G;\mathbb F_q)$ vanish for every $n>0$, as continuous cohomology does? | Is this 0 for n > 0, as it is for Hncts ? Cohomology of Finite Groups: Interactions and Applications 551 | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,650 | OWR-14299915-006 | Noetherian Twisted Cohomology of Finite $p$-Groups | For every finite $p$-group $G$, prove that its permutation-twisted cohomology ring $H^{\bullet\bullet}(G)$ is Noetherian. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 4 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-006",
"literature-status:open",
"owr-year:2026"
] | Noetherianity of the twisted cohomology ring for every finite p-group is still posed as a conjecture; current work extends the construction and proves control results but not the required Noetherianity. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2509.00954",
"label": "Miller, Permutation twisted cohomology, remixed"
},
{
"url": "https://arxiv.org/abs/2307.04398",
"label": "Balmer--Gallauer, The tt-geome... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | For every finite $p$-group $G$, prove that its permutation-twisted cohomology ring $H^{\bullet\bullet}(G)$ is Noetherian. | Conjecture 9. Let G be a finite p-group. Then H•• (G) is Noetherian. Independent of Noetherianity, we obtain that the above theorem holds at the trivial subgroup of G. As a result, we obtain a nice alternative characterization of the usual cohomology ring of a p-group. In particular, Spc(K(G)) is a Dirac scheme. (2) The... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,651 | OWR-14299915-007 | Periodic Loci for Derived Permutation Modules | Let $G$ be a finite or profinite group and let $H\leq G$ be a finite or profinite $p$-subgroup. Prove that the point $P(H,p)$ in the spectrum of compact derived permutation modules is periodic if and only if $p$ is periodic; equivalently, prove that the periodic locus is the complement of the closed points. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-007",
"literature-status:partially-solved",
"owr-year:2026"
] | The periodic-locus conjecture is proved when the ambient group is a pro-p group and in important elementary-abelian cases, but the general profinite-group statement remains open because the needed ampleness is not known. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2509.00954",
"label": "Miller, Permutation twisted cohomology, remixed"
},
{
"url": "https://arxiv.org/abs/2307.04398",
"label": "Balmer--Gallauer, The tt-geome... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | Let $G$ be a finite or profinite group and let $H\leq G$ be a finite or profinite $p$-subgroup. Prove that the point $P(H,p)$ in the spectrum of compact derived permutation modules is periodic if and only if $p$ is periodic; equivalently, prove that the periodic locus is the complement of the closed points. | Conjecture 7. Let G be a (pro-)finite group, H \leq G a (pro-)p-subgroup. Then P(H, p) is periodic iff p is. In other words, the periodic locus of Spec(DPerm(G; k)\omega ) is precisely the com- plement of the closed points. Remark 8 (Twisted group cohomology). The natural proof strategy would follow the one of Example 2 ... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,652 | OWR-14299915-008 | $E_\infty$ Structures from Frobenius Lifts | For a flat spectrum $A$, characterize which commutative ring structures on $\pi_0(A)$ equipped with Frobenius lifts $\varphi_p:\pi_0(A)\to\pi_0(A)^\wedge_p$ arise from $E_\infty$ ring structures, and determine whether such extensions are unique. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-008",
"literature-status:partially-solved",
"owr-year:2026"
] | For flat spectra, the report proves that H-infinity structures are determined by the commutative ring on pi-zero and the Frobenius lifts, but it explicitly leaves extension and uniqueness of E-infinity structures unresolved. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://arxiv.org/abs/2405.06448",
"label": "Carmeli--Nikolaus--Yuan, Maps between spherical group rings"
},
{
"url": "https://doi.org/10.4310/ACTA.2018.v221.n2.a1",
"label": "Nikola... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | For a flat spectrum $A$, characterize which commutative ring structures on $\pi_0(A)$ equipped with Frobenius lifts $\varphi_p:\pi_0(A)\to\pi_0(A)^\wedge_p$ arise from $E_\infty$ ring structures, and determine whether such extensions are unique. | So a natural question is the following: Question 8. How much of the E\infty ring structure is determined by the Frobenius lifts ϕp : \pi0 (A) \to \pi0 (A)∧ p? \pi0 map(X ⊗n , X)hΣn , which satisfy composition relations only up to homotopy. The following observation goes back to Nikolaus-Scholze [2], and is an important... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,653 | OWR-14299915-009 | Rank Conjecture for Free Elementary-Abelian Actions | If $G=(\mathbb Z/p)^r$ acts freely on a finite-dimensional CW complex homotopy equivalent to a product of $k$ spheres, prove that $r\leq k$. In particular, settle the remaining equidimensional cases $p=2$ with sphere dimension $3$ or $7$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
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"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-009",
"literature-status:partially-solved",
"owr-year:2026"
] | The rank bound is established in broad cases, including all odd primes in the cited equidimensional setting, while the full generality and the remaining p-equals-two cases n-equals-three and seven are still open. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "originating Oberwolfach report"
},
{
"url": "https://eudml.org/doc/142960",
"label": "Carlsson, On the Rank of Abelian Groups Acting Freely on Products of Spheres"
},
{
"url": "https://doi.org/10.1016/0040-9383(88)90024-9",
"la... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | If $G=(\mathbb Z/p)^r$ acts freely on a finite-dimensional CW complex homotopy equivalent to a product of $k$ spheres, prove that $r\leq k$. In particular, settle the remaining equidimensional cases $p=2$ with sphere dimension $3$ or $7$. | The rank conjecture is still open in its full generality. For free actions of G = (Z/p)r on X ≃ (S n )k , the conjecture is completely settled except for the cases p = 2 and n = 3, 7. Cohomology of Finite Groups: Interactions and Applications 567 | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,654 | OWR-14299915-015 | Synthetic-Building Classification Preamble | What do they look like? How many of them are there? What are the minimal synthetic buildings? Cohomology of Finite Groups: Interactions and Applications 583 | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
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"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | Unrecoverable OWR statement (The official report confirms that these broad questions introduce the precise minimal-synthetic-building conjectures preserved in the follow) | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-015",
"literature-status:partially-solved",
"owr-year:2026"
] | Synthetic buildings now have a general definition, many constructions, and partial classification results, but the questions of their full form and number remain open; current work still states classification conjectures. | [
{
"url": "https://arxiv.org/abs/2608.13414",
"label": "Gullerud--Webb, Synthetic Buildings for Finite Groups"
},
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Oberwolfach Report 9/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | null | What do they look like? How many of them are there? What are the minimal synthetic buildings? Cohomology of Finite Groups: Interactions and Applications 583 | unrecoverable | The official report confirms that these broad questions introduce the precise minimal-synthetic-building conjectures preserved in the following record. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,655 | OWR-14299915-016 | Classification of Minimal Synthetic Buildings | For each finite group $G$ and prime $p$, prove that there are only finitely many equivariant homotopy types of minimal synthetic buildings; prove that the $p$-subgroup complex $\Delta(\mathcal S_p(G))$ has the equivariant homotopy type of one; and prove that every minimal synthetic building is equivariantly homotopy eq... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-016",
"literature-status:partially-solved",
"owr-year:2026"
] | The rank-at-most-two case of Conjecture 7 is proved, while Conjectures 5 and 6 and the general Lie-rank classification remain unresolved. | [
{
"url": "https://arxiv.org/abs/2608.13414",
"label": "Gullerud--Webb, Synthetic Buildings for Finite Groups"
},
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Oberwolfach Report 9/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | For each finite group $G$ and prime $p$, prove that there are only finitely many equivariant homotopy types of minimal synthetic buildings; prove that the $p$-subgroup complex $\Delta(\mathcal S_p(G))$ has the equivariant homotopy type of one; and prove that every minimal synthetic building is equivariantly homotopy eq... | Conjecture 5. For each finite group and prime p, the p-subgroups complex ∆(Sp (G)) always has the equivariant homotopy type of a minimal synthetic build- ing. Conjecture 6. Minimal synthetic buildings are all equivariantly homotopy equiv- alent to homomorphic images of ∆(Sp (G)). Conjecture 7. For groups of Lie type in ... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,656 | OWR-14299915-017 | Duplicate Noncompact Casacuberta-Dicks Question | Can we say whether X G , the fixed point set, is non-empty? Group actions on compact contractible complexes Kevin Ivan Piterman (joint work with Ivan Sadofschi Costa) | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
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"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | Duplicate/merged OWR extraction (The official report states that the compact contractible two-dimensional case is solved and leaves exactly the noncompact Casacuberta-Dicks ) | [
"OWR-14298007-009"
] | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-017",
"literature-status:partially-solved",
"owr-year:2026"
] | The unrestricted fixed-point question has negative examples, including finite fixed-point-free actions on compact contractible complexes, while the compact contractible 2-complex case is settled positively and broader cases remain open. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Piterman--Sadofschi Costa, Oberwolfach Report 9/2026"
},
{
"url": "https://arxiv.org/abs/2102.11458",
"label": "Sadofschi Costa, Group actions on contractible 2-complexes I"
},
{
"url": "https://arxiv.org/abs/2102.11459",
"labe... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | null | Can we say whether X G , the fixed point set, is non-empty? Group actions on compact contractible complexes Kevin Ivan Piterman (joint work with Ivan Sadofschi Costa) | unrecoverable | The official report states that the compact contractible two-dimensional case is solved and leaves exactly the noncompact Casacuberta-Dicks question already preserved as OWR-14298007-009. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,657 | OWR-14299915-018 | Recovering Fixed-Point Homology in Smith Theory | For a finite $\mathbb Z/p$-CW complex $X$, determine how much of the homology of the fixed-point space $X^{\mathbb Z/p}$ can be reconstructed from the homological and equivariant-cohomological data of $X$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-018",
"literature-status:partially-solved",
"owr-year:2026"
] | Smith localization and the unstable-module description recover substantial fixed-point cohomology from Borel data, and recent structured work recovers the p-adic homotopy type in the finite CW setting, but no complete homological reconstruction is known. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Subramanian--Burklund, Smith theory revisited"
},
{
"url": "https://web.math.ku.dk/~jg/students/subramanian-phdthesis.pdf",
"label": "Subramanian, Categorification of Smith Theory"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | For a finite $\mathbb Z/p$-CW complex $X$, determine how much of the homology of the fixed-point space $X^{\mathbb Z/p}$ can be reconstructed from the homological and equivariant-cohomological data of $X$. | How much of the homological information about X Z/p can be re- covered from that of X? The report passage is retained verbatim as the relevant context for this extracted problem. | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,658 | OWR-14299915-021 | Vanishing for Non-$p$-Nilpotent Compact Lie Groups | Let $G$ be a positive-dimensional compact Lie group such that $\pi_0(G)$ is not $p$-nilpotent. Is the map $\pi_*(\Gamma_k C_*(\Omega BG^\wedge_p))\to\pi_*(C_*(\Omega BG^\wedge_p))$ zero? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-021",
"literature-status:open",
"owr-year:2026"
] | The stated compact-Lie-group vanishing question is posed as an open problem; the finite-group analogue is known, but no proof or counterexample for the stated general case was found. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Benson, The Omega-Tate spectrum, Oberwolfach Report 9/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | Let $G$ be a positive-dimensional compact Lie group such that $\pi_0(G)$ is not $p$-nilpotent. Is the map $\pi_*(\Gamma_k C_*(\Omega BG^\wedge_p))\to\pi_*(C_*(\Omega BG^\wedge_p))$ zero? | Question 2.2. If G is a compact Lie group of dimension d > 0 with \pi0 (G) not p-nilpotent, does it follow that the map \pi∗ (Γk C∗ (ΩBGp )) −\to \pi∗ (C∗ (ΩBGp )) is zero? 2.4. The singularity category as a bounded derived category. The main result is that the singularity category is the bounded derived category of th... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,659 | OWR-14299915-022 | Free Finite-Group Actions on Products of Spheres | Prove that a finite group $G$ acts freely on a finite CW complex homotopy equivalent to a product of $r$ spheres if and only if $G$ has no elementary abelian subgroup of rank $r+1$. In particular, determine whether $Qd(p)$ acts freely on a finite complex homotopy equivalent to $S^n\times S^m$ for odd $p$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | partially_solved | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-022",
"literature-status:partially-solved",
"owr-year:2026"
] | The rank-one case and substantial rank-two families are solved, but the Benson--Carlson characterization still reduces to unresolved free actions for Qd(p) at odd primes. | [
{
"url": "https://arxiv.org/abs/math/0010096",
"label": "Adem--Smith, Periodic Complexes and Group Actions"
},
{
"url": "https://arxiv.org/abs/math/0503746",
"label": "Jackson, Qd(p)-free rank two groups"
},
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Oberwolfach Report ... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | Prove that a finite group $G$ acts freely on a finite CW complex homotopy equivalent to a product of $r$ spheres if and only if $G$ has no elementary abelian subgroup of rank $r+1$. In particular, determine whether $Qd(p)$ acts freely on a finite complex homotopy equivalent to $S^n\times S^m$ for odd $p$. | A fundamental question in the theory of transformation groups is: Which finite groups can act freely on a given product of spheres? Free symmetries of products of spheres and Steenrod closed parameter ideals Marc Stephan (joint work with Henrik Rüping, Ergün Yalçın) | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,660 | OWR-14299915-023 | Steenrod-Closed Parameter Ideals at Every Prime | For every prime $p$, classify the Steenrod-closed parameter ideals in $\mathbb F_p[a,b]^{\operatorname{SL}_2(p)}$ and prove that no ideals occur beyond the fibered, twisted, and mixed families identified for $p=3$. | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 4 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-023",
"literature-status:open",
"owr-year:2026"
] | The classification of Steenrod-closed parameter ideals is proved for p=3, but the report explicitly leaves its extension to every prime as a conjecture and no later proof was located. | [
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Stephan--Ruping--Yalcin, Oberwolfach Report 9/2026"
},
{
"url": "https://arxiv.org/abs/2503.14349",
"label": "Ruping--Stephan, Steenrod-closed parameter ideals"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | For every prime $p$, classify the Steenrod-closed parameter ideals in $\mathbb F_p[a,b]^{\operatorname{SL}_2(p)}$ and prove that no ideals occur beyond the fibered, twisted, and mixed families identified for $p=3$. | We conjecture that Theorem 5 extends to every prime p. The report passage is retained verbatim as the relevant context for this extracted problem. | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,661 | OWR-14299915-024 | Diffeomorphic Doubles of Homotopy-Equivalent Two-Complexes | Let $X$ and $Y$ be finite two-dimensional CW complexes with finite fundamental group, and let $M(X)$ and $M(Y)$ be their smooth four-dimensional doubles. If $X\simeq Y$, must $M(X)$ and $M(Y)$ be diffeomorphic? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-024",
"literature-status:open",
"owr-year:2026"
] | The question whether homotopy-equivalent 2-complexes have diffeomorphic doubles remains unresolved; the quadratic bias invariant gives new obstructions and examples but not a general diffeomorphism theorem. | [
{
"url": "https://doi.org/10.1016/j.aim.2025.110728",
"label": "Hambleton--Nicholson, Four-manifolds, two-complexes and the quadratic bias invariant"
},
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Oberwolfach Report 9/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | Let $X$ and $Y$ be finite two-dimensional CW complexes with finite fundamental group, and let $M(X)$ and $M(Y)$ be their smooth four-dimensional doubles. If $X\simeq Y$, must $M(X)$ and $M(Y)$ be diffeomorphic? | More generally, we ask: Question 2.2. 2. The quadratic bias invariant and Tate cohomology The quadratic bias invariant \betaQ : M 4 (G) \to BQ (G) is defined for a thickening M (X) by an invariant in the isometry group of the hyperbolic Tate form ebG : (( b L)∗ ⊕ b L) \times (( b L)∗ ⊕ b L) \to Z/|G| associated to the e... | corrected_verified | The statement was reconstructed and checked against the official Oberwolfach report. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,662 | OWR-14299915-025 | Exotic-Four-Sphere Special-Case Discussion | In this case we have X ≃ Y , M (X) = S 4 , M (Y ) ≃ S 4 and so 2.2 is equivalent to the question of whether M (Y ) is an exotic 4-sphere. Cohomology of Finite Groups: Interactions and Applications 605 | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610](https://doi.org/10.4171/owr/2026/9)
## Literature review (chec... | 3 | open | null | 2,026 | 7 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | Unrecoverable OWR statement (The official report identifies this fragment as a special case of the preceding doubles question rather than a separate extracted problem. N) | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299915-025",
"literature-status:open",
"owr-year:2026"
] | The special case is equivalent to the existence question for an exotic 4-sphere, which remains unresolved; the cited doubling work supplies context but no resolution. | [
{
"url": "https://doi.org/10.1016/j.aim.2025.110728",
"label": "Hambleton--Nicholson, Four-manifolds, two-complexes and the quadratic bias invariant"
},
{
"url": "https://doi.org/10.4171/owr/2026/9",
"label": "Oberwolfach Report 9/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/9 | null | null | null | null | In this case we have X ≃ Y , M (X) = S 4 , M (Y ) ≃ S 4 and so 2.2 is equivalent to the question of whether M (Y ) is an exotic 4-sphere. Cohomology of Finite Groups: Interactions and Applications 605 | unrecoverable | The official report identifies this fragment as a special case of the preceding doubles question rather than a separate extracted problem. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. | null | null | null | null | null | Cohomology of Finite Groups: Interactions and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/9, pp. 541--610 |
30,006,663 | OWR-14299916-001 | Complexity and Generalization in Overparameterized Learning | Which intrinsic notions of complexity govern learning and generalization in highly and moderately overparameterized models? Why do gradient-based optimization methods consistently select solutions that generalize well? How do useful representations emerge during training, and what are the computational limitations of m... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | open | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
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"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299916-001",
"literature-status:open",
"owr-year:2026"
] | The listed questions about complexity, implicit optimization bias, representation emergence, and computational limits are broad research problems with substantial partial results but no unified resolution. | [
{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | Which intrinsic notions of complexity govern learning and generalization in highly and moderately overparameterized models? Why do gradient-based optimization methods consistently select solutions that generalize well? How do useful representations emerge during training, and what are the computational limitations of m... | What notions of complexity govern learning and generalization in highly overparameterized and moderately overparametrized models? Why do gradient- based optimization methods consistently identify solutions that generalize well? How do useful representations emerge during training? What are the computa- tional limitatio... | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,664 | OWR-14299916-002 | Unified Mathematical Theory of Modern Learning | Can the geometric, probabilistic, computational, representational, and dynamical perspectives on modern learning systems be integrated into a coherent unified mathematical theory? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | open | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
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"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | [
"OWR-14299916-011"
] | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299916-002",
"literature-status:open",
"owr-year:2026"
] | The possibility of a unified mathematical theory synthesizing the listed modern learning phenomena is explicitly left open in the workshop report, with no subsequent resolution located. | [
{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | Can the geometric, probabilistic, computational, representational, and dynamical perspectives on modern learning systems be integrated into a coherent unified mathematical theory? | Whether such a synthesis ultimately takes the form of a unified mathematical theory remains an open question. Modern and Emerging Phenomena in Machine Learning 689 Modern and Emerging Phenomena in Machine Learning 691 | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,665 | OWR-14299916-003 | Curricula for Algorithmic Reasoning in Sequence Models | Beyond currently studied stylized settings, which curricula encourage algorithmic reasoning in transformers? How can disruptive but correctly labeled examples be identified for structured state-space or hybrid architectures? Can these insights be converted into principled data-selection or curriculum-learning methods f... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | open | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
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"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"id": 15,
"name": "oberwolfach_reports_open_problems",
"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299916-003",
"literature-status:open",
"owr-year:2026"
] | Existing Transformer and state-space-model studies cover stylized curricula and architectures, but no general characterization of curricula or disruptive hybrid examples answering the stated questions is known. | [
{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | Beyond currently studied stylized settings, which curricula encourage algorithmic reasoning in transformers? How can disruptive but correctly labeled examples be identified for structured state-space or hybrid architectures? Can these insights be converted into principled data-selection or curriculum-learning methods f... | Can one characterize, beyond the stylized settings studied so far, which curricula encourage algorithmic reasoning in Transformers? How should one identify potentially disruptive but clean examples for SSMs or hy- brid architectures? Modern and Emerging Phenomena in Machine Learning 695 | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,666 | OWR-14299916-004 | Cross-Entropy Dynamics Beyond Hadamard Initialization | For the multiclass two-layer linear unconstrained-features model with balanced standard-basis inputs, weights $W_t\in\mathbb R^{k\times d}$ and $H_t\in\mathbb R^{d\times k}$, and cross-entropy loss, determine the gradient-flow dynamics under general initialization. Do the normalized parameters converge to neural-collap... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | partially_solved | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
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"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299916-004",
"literature-status:partially-solved",
"owr-year:2026"
] | For the canonical balanced two-layer linear model, recent work proves global convergence of cross-entropy gradient flow to neural-collapse geometry and gives a tractable trajectory under Hadamard initialization, while dependence on general initialization and class count remains unresolved. | [
{
"url": "https://arxiv.org/abs/2512.04006",
"label": "Garrod--Keating--Thrampoulidis, Diagonalizing the Softmax"
},
{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | For the multiclass two-layer linear unconstrained-features model with balanced standard-basis inputs, weights $W_t\in\mathbb R^{k\times d}$ and $H_t\in\mathbb R^{d\times k}$, and cross-entropy loss, determine the gradient-flow dynamics under general initialization. Do the normalized parameters converge to neural-collap... | Three ques- tions drove the talk: (1) What do Wt and Ht converge to? (2) What does the trajectory look like – is convergence monotone in some metric? (3) How does the answer depend on initialization and on the number of classes k? Cross-entropy (CE) is the workhorse loss for training deep-net classifiers, yet our theore... | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,667 | OWR-14299916-005 | Inference and Training Effects of the Softmax Bottleneck | When nucleus sampling is used at inference time, do the probabilities of tokens outside the top one thousand have negligible effects on performance? Does the softmax bottleneck materially limit performance during training even when it does not significantly restrict the probabilities of the most likely tokens at infere... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | partially_solved | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
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"name": "L3: Advanced",
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"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"display_name": "Oberwolfach Reports Open Problems",
"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
"oberwolfach-reports",
"OWR-14299916-005",
"literature-status:partially-solved",
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] | The ICLR 2026 analysis gives bounds and experiments for top-m representability and shows that many likely-token distributions are attainable, but it does not characterize all excluded distributions or settle their performance significance. | [
{
"url": "https://openreview.net/forum?id=DgJqQk6y19",
"label": "Basri--Jacobs, The Softmax Bottleneck"
},
{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
}
] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | When nucleus sampling is used at inference time, do the probabilities of tokens outside the top one thousand have negligible effects on performance? Does the softmax bottleneck materially limit performance during training even when it does not significantly restrict the probabilities of the most likely tokens at infere... | A key question is which probability distributions cannot be represented by LLMs due to this bottleneck, and how significant are they to performance? How do the answers to these questions vary with m? The Softmax Bottleneck Does Not Limit the Probabilities of the Most Likely Tokens Ronen Basri Modern and Emerging Phenome... | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,668 | OWR-14299916-006 | Extremal Top-$m$ Representability of Output Projection Matrices | For a vocabulary of $N$ tokens and embedding dimension $d$, what is the largest $m$ for which there exists an output projection matrix $A\in\mathbb R^{N\times d}$ such that, for every set of $m$ tokens, some embedding $x\in\mathbb R^d$ makes those tokens the $m$ most probable under $\operatorname{softmax}(Ax)$? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | partially_solved | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
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"description": "Open and partially solved problems extracted from Oberwolfach Reports workshop discussions, with report provenance and dated literature assessments.",
"slug": "oberwolfach-reports-open... | null | {
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"checked_at": "2026-10-14",
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} | null | null | null | [Discussion #4 literature review, 2026-10-14] **partially solved (the largest m is ⌊d/2⌋ under the source paper's hypothesis N ≥ 3m + 1)**; the dataset currently labels it open. R. Basri (with D. W. Jacobs), OWR 11/2026, p. 707: for a vocabulary of N tokens and embedding dimension d, what is the largest m for which som... | [
"owr",
"oberwolfach-reports",
"OWR-14299916-006",
"literature-status:open",
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] | The ICLR 2026 paper derives sign-rank bounds and empirical values for the largest top-m regime, but leaves the exact extremal m in the stated OPM problem undetermined. | [
{
"url": "https://openreview.net/forum?id=DgJqQk6y19",
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{
"url": "https://doi.org/10.4171/owr/2026/11",
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | For a vocabulary of $N$ tokens and embedding dimension $d$, what is the largest $m$ for which there exists an output projection matrix $A\in\mathbb R^{N\times d}$ such that, for every set of $m$ tokens, some embedding $x\in\mathbb R^d$ makes those tokens the $m$ most probable under $\operatorname{softmax}(Ax)$? | We conjecture, and support with empirical evidence, that at inference time, the probabilities of tokens beyond the top 1,000 will not significantly affect performance when nucleus sampling is used. What is the largest m for which there exists an OPM such that, for any collection of m tokens, there exists an embedding tha... | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,669 | OWR-14299916-007 | Human Alignment of Artificial Intelligence Systems | How can artificial intelligence systems be aligned more closely with human behavior and uncertainty, particularly by representing the full fine-to-coarse hierarchy of human conceptual knowledge? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | partially_solved | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
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"slug": "oberwolfach-reports-open... | [
"OWR-14299916-008"
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"owr",
"oberwolfach-reports",
"OWR-14299916-007",
"literature-status:partially-solved",
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] | Human-aligned visual representations have been constructed and shown to improve human-behavioral similarity, uncertainty, and robustness, but general alignment of AI behavior with human users remains unresolved. | [
{
"url": "https://arxiv.org/abs/2409.06509",
"label": "Muttenthaler et al., Aligning Machine and Human Visual Representations"
},
{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | How can artificial intelligence systems be aligned more closely with human behavior and uncertainty, particularly by representing the full fine-to-coarse hierarchy of human conceptual knowledge? | How can the behavior of AI systems be more closely aligned with that of human users? Deep neural networks have achieved success across a wide range of applications, including as models of human behavior in vision tasks. However, neural network training and human learning differ in fundamental ways, and neural networks o... | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,670 | OWR-14299916-008 | Duplicate Human-Alignment Question | What is missing for modern learning systems to exhibit more human-like behavior? How can the behavior of AI systems be more closely aligned with that of human users? Klaus-Robert Müller Modern and Emerging Phenomena in Machine Learning 713 | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | partially_solved | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
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"slug": "oberwolfach-reports-open... | null | null | Duplicate/merged OWR extraction (This extraction repeats the same human-alignment question retained in OWR-14299916-007 and appends an author name and page footer; it is not) | [
"OWR-14299916-007"
] | null | null | [
"owr",
"oberwolfach-reports",
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] | A demonstrated missing ingredient is multilevel human conceptual structure, and transferring human judgments improves several measures, but this is one task-specific intervention rather than a complete account of human-like learning. | [
{
"url": "https://arxiv.org/abs/2409.06509",
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{
"url": "https://doi.org/10.4171/owr/2026/11",
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | null | What is missing for modern learning systems to exhibit more human-like behavior? How can the behavior of AI systems be more closely aligned with that of human users? Klaus-Robert Müller Modern and Emerging Phenomena in Machine Learning 713 | unrecoverable | This extraction repeats the same human-alignment question retained in OWR-14299916-007 and appends an author name and page footer; it is not a distinct problem. No separate clean statement is promoted, and the original record remains publicly visible for provenance. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,671 | OWR-14299916-009 | Hidden-Layer Depth Complexity of Piecewise-Linear Functions | What is the exact minimum hidden-layer depth required for a ReLU network to represent the function $(x_1,\ldots,x_d)\mapsto\max\{x_1,\ldots,x_d\}$? More generally, does there exist a continuous piecewise-linear function that cannot be represented by any ReLU network with at most two hidden layers? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | open | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
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"slug": "oberwolfach-reports-open... | null | null | null | null | null | null | [
"owr",
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"OWR-14299916-009",
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{
"url": "https://arxiv.org/abs/2607.21651",
"label": "Ruess et al., Shallower ReLU Network Representations"
},
{
"url": "https://arxiv.org/abs/2505.14338",
"label": "Bakaev et al., Better Neural Network Expressivity"
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{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberw... | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | What is the exact minimum hidden-layer depth required for a ReLU network to represent the function $(x_1,\ldots,x_d)\mapsto\max\{x_1,\ldots,x_d\}$? More generally, does there exist a continuous piecewise-linear function that cannot be represented by any ReLU network with at most two hidden layers? | The talk discussed results and open problems around three structural questions about ReLU neural networks: which functions they can represent exactly, how difficult it is to verify properties of a trained network, and to what extent the parameters of a network are identifiable from the function it computes. However, the e... | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,672 | OWR-14299916-010 | New Theoretical Principles for Foundation Models | Do the phenomena observed in contemporary foundation models require fundamentally new mathematical principles, or can they be explained by extensions of existing theories of approximation, optimization, and statistical learning? | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | open | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | true | {
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"owr",
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"OWR-14299916-010",
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{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | Do the phenomena observed in contemporary foundation models require fundamentally new mathematical principles, or can they be explained by extensions of existing theories of approximation, optimization, and statistical learning? | An important open question is whether the phenomena observed in contemporary foundation models require fundamentally new principles or can be understood through extensions of existing theories of approximation, optimization, and statistical learning. Direction 4: Understanding the Computational Limits of Learning. Clas... | corrected_verified | The official Oberwolfach report was checked directly. The clean statement preserves the complete open target while repairing OCR, notation, and source-dependent wording. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,673 | OWR-14299916-011 | Duplicate Unified-Theory Question | The workshop demonstrated both the remarkable progress that has been achieved in understanding modern learning systems and the sub- stantial challenges that remain open. Whether machine learning will ultimately admit such a unified mathematical description remains an open question. Modern and Emerging Phenomena in Machi... | Collected from Oberwolfach Reports, which document mathematical workshops and their open-problem discussions.
**Source report:** [Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736](https://doi.org/10.4171/owr/2026/11)
## Literature review (checked 202... | 3 | open | null | 2,026 | 15 | 15 | 0 | 0 | 2026-08-25T00:00:00 | 2026-09-16T12:06:53 | false | {
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"slug": "oberwolfach-reports-open... | null | null | Duplicate/merged OWR extraction (This concluding rhetorical question repeats the unified-theory problem retained in OWR-14299916-002 and does not introduce a distinct mathem) | [
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] | null | null | [
"owr",
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{
"url": "https://doi.org/10.4171/owr/2026/11",
"label": "Oberwolfach Report 11/2026"
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] | 2026-08-22T00:00:00 | null | https://doi.org/10.4171/owr/2026/11 | null | null | null | null | The workshop demonstrated both the remarkable progress that has been achieved in understanding modern learning systems and the sub- stantial challenges that remain open. Whether machine learning will ultimately admit such a unified mathematical description remains an open question. Modern and Emerging Phenomena in Machi... | unrecoverable | This concluding rhetorical question repeats the unified-theory problem retained in OWR-14299916-002 and does not introduce a distinct mathematical target. No separate clean statement is promoted, and the original record remains publicly visible for provenance. | null | null | null | null | null | Modern and Emerging Phenomena in Machine Learning (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/11, pp. 683--736 |
30,006,674 | EP-2 | Erdős Problem #2: Can the smallest modulus of a covering system be arbitrarily large? | Can the smallest modulus of a covering system be arbitrarily large? | Erdős Problems (#2, https://www.erdosproblems.com/2). Maintainer status: DISPROVED (LEAN). Erdős prize: $1000. OEIS: A160559. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.2). | 4 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
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"slug": "erdos-problems",
"or... | null | {
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30,006,675 | EP-4 | Erdős Problem #4: Number Theory, Primes | Is it true that, for any $C>0$, there are infinitely many $n$ such that\[p_{n+1}-p_n> C\frac{\log\log n\log\log\log\log n}{(\log\log \log n)^2}\log n?\] | Erdős Problems (#4, https://www.erdosproblems.com/4). Maintainer status: PROVED. Erdős prize: $10000. OEIS: A002386. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.4). | 4 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
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"color_class": "text-red-600 bg-red-50 border-red-200"
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30,006,676 | EP-6 | Erdős Problem #6: Let $d_n=p_{n+1}-p_n$. Are there infinitely many $n$ such that $d_n<d_{n+1}<d_{n+2}$? | Let $d_n=p_{n+1}-p_n$. Are there infinitely many $n$ such that $d_n<d_{n+1}<d_{n+2}$? | Erdős Problems (#6, https://www.erdosproblems.com/6). Maintainer status: PROVED (LEAN). Erdős prize: $100. OEIS: A335277. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.6). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
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30,006,677 | EP-7 | Erdős Problem #7: Is there a distinct covering system all of whose moduli are odd? | Is there a distinct covering system all of whose moduli are odd? | Erdős Problems (#7, https://www.erdosproblems.com/7). Maintainer status: VERIFIABLE. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.7). | 3 | open | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
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"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/7",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "verifiable"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"covering-systems"
] | null | null | null | null | https://www.erdosproblems.com/7 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,678 | EP-8 | Erdős Problem #8: Number Theory, Covering Systems | For any finite colouring of the integers is there a covering system all of whose moduli are monochromatic? | Erdős Problems (#8, https://www.erdosproblems.com/8). Maintainer status: DISPROVED (LEAN). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/8",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"covering-systems"
] | null | null | null | null | https://www.erdosproblems.com/8 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,679 | EP-11 | Erdős Problem #11: Is every large odd integer $n$ the sum of a squarefree number and a power of 2? | Is every large odd integer $n$ the sum of a squarefree number and a power of 2? | Erdős Problems (#11, https://www.erdosproblems.com/11). Maintainer status: OPEN. OEIS: A001220, A377587. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.11). | 3 | open | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/11",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-basis"
] | null | null | null | null | https://www.erdosproblems.com/11 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,680 | EP-13 | Erdős Problem #13: Number Theory | Let $A\subseteq \{1,\ldots,N\}$ be such that there are no $a,b,c\in A$ such that $a\mid(b+c)$ and $a<\min(b,c)$. Is it true that $\lvert A\rvert\leq N/3+O(1)$? | Erdős Problems (#13, https://www.erdosproblems.com/13). Maintainer status: PROVED (LEAN). Erdős prize: $100. OEIS: A002264. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.13). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/13",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/13 | Erdős Problems (erdosproblems.com) | $100 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,681 | EP-16 | Erdős Problem #16: Number Theory, Additive Basis | Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$? | Erdős Problems (#16, https://www.erdosproblems.com/16). Maintainer status: DISPROVED (LEAN). OEIS: A006285. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.16). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/16",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-basis",
"primes"
] | null | null | null | null | https://www.erdosproblems.com/16 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,682 | EP-19 | Erdős Problem #19: If $G$ is an edge-disjoint union of $n$ copies of $K_n$ then is $\chi(G)=n$? | If $G$ is an edge-disjoint union of $n$ copies of $K_n$ then is $\chi(G)=n$? | Erdős Problems (#19, https://www.erdosproblems.com/19). Maintainer status: DECIDABLE. Erdős prize: $500. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.19). | 3 | open | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/19",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "decidable"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"chromatic-number"
] | null | null | null | null | https://www.erdosproblems.com/19 | Erdős Problems (erdosproblems.com) | $500 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,683 | EP-21 | Erdős Problem #21: Combinatorics, Intersecting Family | Let $f(n)$ be minimal such that there is an intersecting family $\mathcal{F}$ of sets of size $n$ (so $A\cap B\neq\emptyset$ for all $A,B\in \mathcal{F}$) with $\lvert \mathcal{F}\rvert=f(n)$ such that any set $S$ with $\lvert S\rvert \leq n-1$ is disjoint from at least one $A\in \mathcal{F}$.
Is it true that\[f(n) \l... | Erdős Problems (#21, https://www.erdosproblems.com/21). Maintainer status: PROVED (LEAN). Erdős prize: $500. OEIS: A391599. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.21). | 3 | solved | Paul Erdős | null | 2 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/21",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"combinatorics",
"intersecting-family"
] | null | null | null | null | https://www.erdosproblems.com/21 | Erdős Problems (erdosproblems.com) | $500 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,684 | EP-22 | Erdős Problem #22: Graph Theory | Let $\epsilon>0$ and $n$ be sufficiently large depending on $\epsilon$. Is there a graph on $n$ vertices with $\geq n^2/8$ many edges which contains no $K_4$ such that the largest independent set has size at most $\epsilon n$? | Erdős Problems (#22, https://www.erdosproblems.com/22). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.22). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/22",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"graph-theory"
] | null | null | null | null | https://www.erdosproblems.com/22 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,685 | EP-23 | Erdős Problem #23: Graph Theory | Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges? | Erdős Problems (#23, https://www.erdosproblems.com/23). Maintainer status: FALSIFIABLE. OEIS: A389646. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.23). | 3 | open | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/23",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "falsifiable"
} | null | null | null | null | [
"erdos-problems",
"graph-theory"
] | null | null | null | null | https://www.erdosproblems.com/23 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,686 | EP-24 | Erdős Problem #24: Graph Theory | Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$? | Erdős Problems (#24, https://www.erdosproblems.com/24). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.24). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/24",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"graph-theory"
] | null | null | null | null | https://www.erdosproblems.com/24 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,687 | EP-26 | Erdős Problem #26: Number Theory, Divisors | Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$? | Erdős Problems (#26, https://www.erdosproblems.com/26). Maintainer status: DISPROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.26). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/26",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"divisors"
] | null | null | null | null | https://www.erdosproblems.com/26 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,688 | EP-27 | Erdős Problem #27: Number Theory, Covering Systems | An $\epsilon$-almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\cdots<n_k$ such that the density of those integers which satisfy none of them is $\leq \epsilon$.
Is there a constant $C>1$ such that for every $\epsilon>0$ and $N\geq 1$ there is an $\epsilon$-almost covering system... | Erdős Problems (#27, https://www.erdosproblems.com/27). Maintainer status: DISPROVED. Erdős prize: $100. | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/27",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"covering-systems"
] | null | null | null | null | https://www.erdosproblems.com/27 | Erdős Problems (erdosproblems.com) | $100 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,689 | EP-29 | Erdős Problem #29: Number Theory, Additive Basis | Is there an explicit construction of a set $A\subseteq \mathbb{N}$ such that $A+A=\mathbb{N}$ but $1_A\ast 1_A(n)=o(n^\epsilon)$ for every $\epsilon>0$? | Erdős Problems (#29, https://www.erdosproblems.com/29). Maintainer status: PROVED (LEAN). Erdős prize: $100. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.29). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/29",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-basis"
] | null | null | null | null | https://www.erdosproblems.com/29 | Erdős Problems (erdosproblems.com) | $100 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,690 | EP-31 | Erdős Problem #31: Number Theory, Additive Basis | Given any infinite set $A\subset \mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ contains all except finitely many integers. | Erdős Problems (#31, https://www.erdosproblems.com/31). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.31). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/31",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-basis"
] | null | null | null | null | https://www.erdosproblems.com/31 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,691 | EP-34 | Erdős Problem #34: Number Theory | For any permutation $\pi\in S_n$ of $\{1,\ldots,n\}$ let $S(\pi)$ count the number of distinct consecutive sums, that is, sums of the shape $\sum_{u\leq i\leq v}\pi(i)$. Is it true that\[S(\pi) = o(n^2)\]for all $\pi\in S_n$? | Erdős Problems (#34, https://www.erdosproblems.com/34). Maintainer status: DISPROVED (LEAN). OEIS: A389241, A234813, A390187. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.34). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/34",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/34 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,692 | EP-35 | Erdős Problem #35: Number Theory, Additive Basis | Let $B\subseteq\mathbb{N}$ be an additive basis of order $k$ with $0\in B$. Is it true that for every $A\subseteq\mathbb{N}$ we have\[d_s(A+B)\geq \alpha+\frac{\alpha(1-\alpha)}{k},\]where $\alpha=d_s(A)$ and\[d_s(A) = \inf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N}\]is the Schnirelmann density? | Erdős Problems (#35, https://www.erdosproblems.com/35). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.35). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/35",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-basis"
] | null | null | null | null | https://www.erdosproblems.com/35 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,693 | EP-37 | Erdős Problem #37: Number Theory, Additive Combinatorics | We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density.
Can a lacunary set $A\subset\mathbb{N}$ be an essential component? | Erdős Problems (#37, https://www.erdosproblems.com/37). Maintainer status: DISPROVED (LEAN). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/37",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-combinatorics"
] | null | null | null | null | https://www.erdosproblems.com/37 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,694 | EP-45 | Erdős Problem #45: Number Theory, Unit Fractions | Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k: d\mid n_k\}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$? | Erdős Problems (#45, https://www.erdosproblems.com/45). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.45). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/45",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"unit-fractions",
"ramsey-theory"
] | null | null | null | null | https://www.erdosproblems.com/45 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,695 | EP-46 | Erdős Problem #46: Number Theory, Unit Fractions | Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$? | Erdős Problems (#46, https://www.erdosproblems.com/46). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.46). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/46",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"unit-fractions",
"ramsey-theory"
] | null | null | null | null | https://www.erdosproblems.com/46 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,696 | EP-47 | Erdős Problem #47: Number Theory, Unit Fractions | If $\delta>0$ and $N$ is sufficiently large in terms of $\delta$, and $A\subseteq\{1,\ldots,N\}$ is such that $\sum_{a\in A}\frac{1}{a}>\delta \log N$ then must there exist $S\subseteq A$ such that $\sum_{n\in S}\frac{1}{n}=1$? | Erdős Problems (#47, https://www.erdosproblems.com/47). Maintainer status: PROVED (LEAN). Erdős prize: $100. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.47). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/47",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"unit-fractions"
] | null | null | null | null | https://www.erdosproblems.com/47 | Erdős Problems (erdosproblems.com) | $100 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,697 | EP-48 | Erdős Problem #48: Are there infinitely many integers $n,m$ such that $\phi(n)=\sigma(m)$? | Are there infinitely many integers $n,m$ such that $\phi(n)=\sigma(m)$? | Erdős Problems (#48, https://www.erdosproblems.com/48). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.48). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/48",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/48 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,698 | EP-49 | Erdős Problem #49: Number Theory, Primes | Let $A=\{a_1<\cdots<a_t\}\subseteq \{1,\ldots,N\}$ be such that $\phi(a_1)<\cdots<\phi(a_t)$. The primes are such an example. Are they the largest possible? Can one show that $\lvert A\rvert<(1+o(1))\pi(N)$ or even $\lvert A\rvert=o(N)$? | Erdős Problems (#49, https://www.erdosproblems.com/49). Maintainer status: PROVED (LEAN). OEIS: A365339, A365474. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.49). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/49",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"primes"
] | null | null | null | null | https://www.erdosproblems.com/49 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,699 | EP-53 | Erdős Problem #53: Number Theory, Additive Combinatorics | Let $A$ be a finite set of integers. Is it true that, for every $k$, if $\lvert A\rvert$ is sufficiently large depending on $k$, then there are least $\lvert A\rvert^k$ many integers which are either the sum or product of distinct elements of $A$? | Erdős Problems (#53, https://www.erdosproblems.com/53). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.53). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/53",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-combinatorics"
] | null | null | null | null | https://www.erdosproblems.com/53 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,700 | EP-54 | Erdős Problem #54: Number Theory, Ramsey Theory | A set of integers $A$ is Ramsey $2$-complete if, whenever $A$ is $2$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$.
Burr and Erdős [BuEr85] showed that there exists a constant $c>0$ such that it cannot be true that\[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2... | Erdős Problems (#54, https://www.erdosproblems.com/54). Maintainer status: SOLVED. Erdős prize: $100. | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/54",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "solved"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"ramsey-theory"
] | null | null | null | null | https://www.erdosproblems.com/54 | Erdős Problems (erdosproblems.com) | $100 | [
"[BuEr85] Burr, S. A. and Erdős, P., A {R}amsey-type property in additive number theory. Glasgow Math. J. (1985), 5--10."
] | null | null | null | null | null | null | null | null | null | null |
30,006,701 | EP-55 | Erdős Problem #55: Number Theory, Ramsey Theory | A set of integers $A$ is Ramsey $r$-complete if, whenever $A$ is $r$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$. Prove any non-trivial bounds about the growth rate of such an $A$ for $r>2$. | Erdős Problems (#55, https://www.erdosproblems.com/55). Maintainer status: SOLVED. Erdős prize: $250. | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/55",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "solved"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"ramsey-theory"
] | null | null | null | null | https://www.erdosproblems.com/55 | Erdős Problems (erdosproblems.com) | $250 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,702 | EP-56 | Erdős Problem #56: Number Theory, Intersecting Family | Let $N\geq p_k$ where $p_k$ is the $k$th prime. Suppose $A\subseteq \{1,\ldots,N\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set? | Erdős Problems (#56, https://www.erdosproblems.com/56). Maintainer status: DISPROVED (LEAN). Erdős prize: $10. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.56). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/56",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"intersecting-family"
] | null | null | null | null | https://www.erdosproblems.com/56 | Erdős Problems (erdosproblems.com) | $10 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,703 | EP-57 | Erdős Problem #57: Graph Theory, Chromatic Number | If $G$ is a graph with infinite chromatic number and $a_1<a_2<\cdots $ are lengths of the odd cycles of $G$ then $\sum \frac{1}{a_i}=\infty$. | Erdős Problems (#57, https://www.erdosproblems.com/57). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.57). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/57",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"chromatic-number",
"cycles"
] | null | null | null | null | https://www.erdosproblems.com/57 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,704 | EP-58 | Erdős Problem #58: Graph Theory, Chromatic Number | If $G$ is a graph which contains odd cycles of $\leq k$ different lengths then $\chi(G)\leq 2k+2$, with equality if and only if $G$ contains $K_{2k+2}$. | Erdős Problems (#58, https://www.erdosproblems.com/58). Maintainer status: PROVED. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.58). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/58",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"chromatic-number",
"cycles"
] | null | null | null | null | https://www.erdosproblems.com/58 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,705 | EP-59 | Erdős Problem #59: Graph Theory, Turan Number | Is it true that the number of graphs on $n$ vertices which do not contain $G$ is\[\leq 2^{(1+o(1))\mathrm{ex}(n;G)}?\] | Erdős Problems (#59, https://www.erdosproblems.com/59). Maintainer status: DISPROVED (LEAN). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/59",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"turan-number"
] | null | null | null | null | https://www.erdosproblems.com/59 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,706 | EP-63 | Erdős Problem #63: Graph Theory, Chromatic Number | Does every graph with infinite chromatic number contain a cycle of length $2^n$ for infinitely many $n$? | Erdős Problems (#63, https://www.erdosproblems.com/63). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.63). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/63",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"chromatic-number",
"cycles"
] | null | null | null | null | https://www.erdosproblems.com/63 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,707 | EP-64 | Erdős Problem #64: Graph Theory, Cycles | Does every finite graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$? | Erdős Problems (#64, https://www.erdosproblems.com/64). Maintainer status: FALSIFIABLE. Erdős prize: $1000. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.64). | 4 | open | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/64",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "falsifiable"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"cycles"
] | null | null | null | null | https://www.erdosproblems.com/64 | Erdős Problems (erdosproblems.com) | $1000 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,708 | EP-67 | Erdős Problem #67: Discrepancy | If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that\[\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?\] | Erdős Problems (#67, https://www.erdosproblems.com/67). Maintainer status: PROVED. Erdős prize: $500. Erdős discrepancy problem OEIS: A181740, A237695. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.67). | 3 | solved | Paul Erdős | null | 8 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/67",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | Erdős discrepancy problem | [
"erdos-problems",
"discrepancy"
] | null | null | null | null | https://www.erdosproblems.com/67 | Erdős Problems (erdosproblems.com) | $500 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,709 | EP-69 | Erdős Problem #69: Number Theory, Irrationality | Is\[\sum_{n\geq 2}\frac{\omega(n)}{2^n}\]irrational? (Here $\omega(n)$ counts the number of distinct prime divisors of $n$.) | Erdős Problems (#69, https://www.erdosproblems.com/69). Maintainer status: PROVED. OEIS: A262153. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.69). | 3 | solved | Paul Erdős | null | 1 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/69",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"irrationality"
] | null | null | null | null | https://www.erdosproblems.com/69 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,710 | EP-71 | Erdős Problem #71: Graph Theory, Cycles | Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$? | Erdős Problems (#71, https://www.erdosproblems.com/71). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.71). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/71",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"cycles"
] | null | null | null | null | https://www.erdosproblems.com/71 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,711 | EP-72 | Erdős Problem #72: Graph Theory, Cycles | Is there a set $A\subset \mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on sufficiently many vertices with average degree $\geq c$ contains a cycle whose length is in $A$? | Erdős Problems (#72, https://www.erdosproblems.com/72). Maintainer status: PROVED. Erdős prize: $100. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.72). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/72",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"cycles"
] | null | null | null | null | https://www.erdosproblems.com/72 | Erdős Problems (erdosproblems.com) | $100 | [] | null | null | null | null | null | null | null | null | null | null |
30,006,712 | EP-73 | Erdős Problem #73: Graph Theory | Let $k\geq 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of size $\geq (n-k)/2$, where $n$ is the number of vertices of $H$. Must $G$ be the union of a bipartite graph and $O_k(1)$ many vertices? | Erdős Problems (#73, https://www.erdosproblems.com/73). Maintainer status: PROVED. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.73). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/73",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"graph-theory"
] | null | null | null | null | https://www.erdosproblems.com/73 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,713 | EP-76 | Erdős Problem #76: Graph Theory, Ramsey Theory | Is it true that in any $2$-colouring of the edges of $K_n$ there must exist at least\[(1+o(1))\frac{n^2}{12}\]many edge-disjoint monochromatic triangles? | Erdős Problems (#76, https://www.erdosproblems.com/76). Maintainer status: PROVED. OEIS: A060407. | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/76",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"ramsey-theory"
] | null | null | null | null | https://www.erdosproblems.com/76 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,714 | EP-79 | Erdős Problem #79: Graph Theory, Ramsey Theory | We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices.
Are there infinitely many graphs $G$ which are not Ramsey size linear but such that all of its subgraphs are? | Erdős Problems (#79, https://www.erdosproblems.com/79). Maintainer status: PROVED. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.79). | 3 | solved | Paul Erdős | null | 3 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/79",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"ramsey-theory"
] | null | null | null | null | https://www.erdosproblems.com/79 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
30,006,715 | EP-83 | Erdős Problem #83: Combinatorics | Suppose that we have a family $\mathcal{F}$ of subsets of $[4n]$ such that $\lvert A\rvert=2n$ for all $A\in\mathcal{F}$ and for every $A,B\in \mathcal{F}$ we have $\lvert A\cap B\rvert \geq 2$. Then\[\lvert \mathcal{F}\rvert \leq \frac{1}{2}\left(\binom{4n}{2n}-\binom{2n}{n}^2\right).\] | Erdős Problems (#83, https://www.erdosproblems.com/83). Maintainer status: PROVED (LEAN). Erdős prize: $500. OEIS: A071799, A387635. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.83). | 3 | solved | Paul Erdős | null | 2 | 8 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/83",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"combinatorics"
] | null | null | null | null | https://www.erdosproblems.com/83 | Erdős Problems (erdosproblems.com) | $500 | [] | null | null | null | null | null | null | null | null | null | null |
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