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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
800,770 | AMR-140-0005 | Positively Curved Projective Varieties — Problem 5 | Let $(X, g)$ be a compact K\"ahler manifold
with semi-positive $($or semi-negative$)$ holomorphic sectional curvature.
Let $V \subset T_X$ be the set of ``appropriate" truly flat tangent vectors.
$\bullet$ Does $V$ determine an integrable subbundle of $T_X$?
$\bullet$ Can we find a subbundle $W \subset T_X$ such that... | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Problem 5
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,771 | AMR-140-0006 | Positively Curved Projective Varieties — Problem 6 | [cf.{[Mat18b, Problem 5.3]}]
\
$\bullet$ Does the blow-up of Hirzebruch surfaces admit
a positive holomorphic sectional curvature?
$\bullet$ Can we classify all the varieties of fixed dimension $($e.g., dimension two or three$)$
with semi-positive $($or positive$)$ holomorphic sectional curvature?
$\bullet$ Can we ... | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Problem 6
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,772 | AMR-140-0007 | Positively Curved Projective Varieties — Problem 7 | Let $X$ be a smooth projective variety with pseudo-effective tangent bundle.
Then is the MRC fibration $\phi : X \to Y$ locally trivial? | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Problem 7
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,773 | AMR-140-0008 | Positively Curved Projective Varieties — Problem 8 | The second difference concerns the positivity of fibers $F$ of $\phi:X \to Y$.
It was proved in [DPS94] that rationally connected manifolds with nef tangent bundle are always Fano manifolds.
However, the same conclusion does not hold for pseudo-effective tangent bundles,
since the tangent bundle of the Hirzebruch surfa... | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Problem 8
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,774 | AMR-140-0009 | Positively Curved Projective Varieties — Problem 9 | Let $X$ be a compact K\"ahler manifold with pseudo-effective tangent bundle.
After we replace $X$ with an appropriate finite \'etale covering of $X$,
we consider the Albanese map $X \to \operatorname{Alb}(X)$.
$\bullet$ Can we prove that the anti-canonical divisor $-K_F$ is big or $F$ is projective?
$\bullet$ Is the ... | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Problem 9
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,775 | AMR-140-0010 | Positively Curved Projective Varieties — Problem 10 | $\bullet$ Does the blow-up of Hirzebruch surfaces at general four points have the pseudo-effective tangent bundle?
\\
$\bullet$ What can we say for the blow-up of Hirzebruch surfaces at special points? | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Problem 10
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,776 | AMR-140-0011 | Positively Curved Projective Varieties — Problem 11 | Under the above assumption for the relative anti-canonical divisor, can we obtain a structure theorem?
Specifically, can we choose a holomorphic morphism $X \to Y'$ that is birationally equivalent to the original $\phi: X \dashrightarrow Y$? | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Problem 11
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,777 | AMR-140-0012 | Positively Curved Projective Varieties — Conjecture 12 | The MRC fibration is defined only by $X$,
and thus it does not reflect information on the boundary $\Delta$.
On the other hand,
the slope rationally connected quotient (sRC quotient for short),
introduced in [Cam16],
gives a generalization of MRC fibrations reflecting the boundary $\Delta$ (see [Cam04, Cam16]).
The fol... | Source list: Matsumura - Open Problems on Structure of Positively Curved Projective Varieties (2022)
Source item: Conjecture 12
Source URL: https://arxiv.org/abs/2211.09153
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Matsumura | 2,022 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,778 | AMR-141-0001 | Local Discrete Holomorphic Dynamics — Open Problem Q1 | *What is the topological structure of (the germ at $p$ of) $K_f$?*
For instance, does $K_f$ have non-empty interior? Is it locally connected at $p$?
Is $K_f\setminus\{p\}$ connected? What is the topological (homological, cohomological) structure
of $U\setminus K_f$? And so on.
\begin{remark}
Both the definition of st... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem Q1
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,779 | AMR-141-0002 | Local Discrete Holomorphic Dynamics — Open Problem Q2 | *What is the dynamical structure of $(K_f,f)$?*
For instance, what is the asymptotic behavior of the orbits? Do
they converge to $p$, or have they a chaotic behavior? Is there a dense
orbit? Do there exist proper $f$-invariant subsets, that is sets
$L\subset K_f$ such that $f(L)\subseteq L$? If they do exist, what is ... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem Q2
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,780 | AMR-141-0003 | Local Discrete Holomorphic Dynamics — Open Problem Q3 | \emph{Find a (possibly small) class $\mathcal{F}$ of holomorphic
local dynamical systems at $O\in\mathbb{C}^n$ such that every
holomorphic local dynamical system $f$ at a point in an
$n$-dimensional complex manifold is holomorphically (respectively,
smoothly, $C^k$, or topologically) locally conjugated to a (possibly) ... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem Q3
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,781 | AMR-141-0004 | Local Discrete Holomorphic Dynamics — Open Problem Q4 | \emph{Find a way to associate a (possibly small) class of
(possibly computable) objects, called *invariants,* to any
holomorphic local dynamical system $f$ at $O\in\mathbb{C}^n$ so
that two holomorphic local dynamical systems at $O$ can be
holomorphically (respectively, smoothly, $C^k$, or topologically) locally conjug... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem Q4
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,782 | AMR-141-0005 | Local Discrete Holomorphic Dynamics — Open Problem Q5 | \emph{Find normal forms and invariants with respect to the relation
of formal conjugacy for holomorphic local dynamical systems at $O\in\mathbb{C}^n$.}
In the rest of this paper we shall describe a few specific open problems,
both in one and in several complex variables; we
refer to [A5,A6] and [Br2] for surveys on th... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem Q5
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,783 | AMR-141-0006 | Local Discrete Holomorphic Dynamics — Open Problem OP1 | \emph{Find families $\{f_{\lambda, a}(z)=\lambda z+z^2 g_a(z)\}_{a\in M}\subset\mathop\mathrm{End}\nolimits(\mathbb{C},O)$ of elliptic discrete holomorphic
dynamical systems such that $f_{\lambda,a}$ is holomorphically linearizable if and only
if $\lambda$ is a Brjuno number.} For instance, *is it true that an elliptic... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP1
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,784 | AMR-141-0007 | Local Discrete Holomorphic Dynamics — Open Problem OP2 | The Brjuno function is clearly quite an irregular function, diverging at a $G_\delta$-dense set
of irrational numbers . A surprising fact is that, on the contrary,
numerical experiments (see, e.g., [Car] and references therein) as well as theoretical
results (see [MMY])
suggest that $\log r+B$ is more regular, even bet... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP2
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,785 | AMR-141-0008 | Local Discrete Holomorphic Dynamics — Open Problem OP3 | \emph{Are the multiplier and the homeomorphism class of the stable set a complete set
of topological invariants for discrete holomorphic local dynamical systems in $\mathop\mathrm{End}\nolimits(\mathbb{C},O)$ having a
Cremer point at the origin?} | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP3
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,786 | AMR-141-0009 | Local Discrete Holomorphic Dynamics — Open Problem OP4 | *Give an effective procedure for deciding whether two germs tangent to the
identity are holomorphically locally conjugated.*
Clearly, a similar question can be asked for parabolic germs; in that case another (topological, formal and holomorphic) invariant is the multiplier, and one has to replace the multiplicity
by a... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP4
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,787 | AMR-141-0010 | Local Discrete Holomorphic Dynamics — Open Problem OP5 | \emph{Let $f\in\mathop\mathrm{End}\nolimits(\mathbb{C}^n,O)$ be non-invertible and dominant. Is it true that $c_\infty=c_\infty(f)$ is an
algebraic integer of order at most $n$, i.e., there are integers $a_0,\ldots, a_{n-1}\in\mathbb{Z}$ such that $c^n_\infty+a_{k-1} c_\infty^{k-1}+\cdots+a_0=0$? Moreover, does there e... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP5
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,788 | AMR-141-0011 | Local Discrete Holomorphic Dynamics — Open Problem OP6 | \emph{Let $f\in\mathop\mathrm{End}\nolimits(\mathbb{C}^n,O)$ be non-invertible and dominant. Is it true that the sequence
$\{c(f^k)\}_{k\in\mathbb{N}}$ satisfies, at least for $k$ large enough, a linear recursion formula
with integer coefficients?} | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP6
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,789 | AMR-141-0012 | Local Discrete Holomorphic Dynamics — Open Problem OP7 | \emph{Let $f\in\mathop\mathrm{End}\nolimits(\mathbb{C}^n,O)$ be a superattracting germ, and let $H$ be the first
non-zero
term in the homogeneous expansion of $f$. Assume that $H$ is degenerate, that is there exists
$v\in\mathbb{C}^n\setminus\{O\}$ such that $H(v)=O$. Is it still true that if the postcritical set of $f... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP7
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,790 | AMR-141-0013 | Local Discrete Holomorphic Dynamics — Open Problem OP9 | \emph{Let $f$, $g\in\mathop\mathrm{End}\nolimits(\mathbb{C}^n,O)$ be two superattracting germs, with $c(f)=c(g)$. Is it true that $f$ and $g$ are
topologically (respectively, holomorphically) locally conjugated if and only if their postcritical sets
are topologically (respectively, holomorphically) conjugated, that is,... | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP9
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,791 | AMR-141-0014 | Local Discrete Holomorphic Dynamics — Open Problem OP10 | \emph{Classify with respect to formal conjugation the superattracting germs
$f\in\mathop\mathrm{End}\nolimits(\mathbb{C}^n,O)$ of order 2 when $n\ge 3$, or of order $3$ when $n=2$.} | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP10
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,792 | AMR-141-0015 | Local Discrete Holomorphic Dynamics — Open Problem OP11 | Of course, one would like to extend these results at least to dimension 3, and thus we can add
two more open problems to the list:
*Classify $3$-dimensional non-invertible dominant rigid germs.* | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP11
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,793 | AMR-141-0016 | Local Discrete Holomorphic Dynamics — Open Problem OP15 | *Find formal normal forms and invariants for germs tangent to the identity
having only degenerate characteristic directions.*
Partial results on this problem can be find in [BM], [AT2] and [AR]. | Source list: Abate - Open Problems in Local Discrete Holomorphic Dynamics (2011)
Source item: Problem OP15
Source URL: https://arxiv.org/abs/1107.4876
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Abate | 2,011 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,794 | AMR-142-0001 | Branko Grunbaum in Many Dimensions — Question 1 | There is still more to do in this area, however. A Venn diagram is said to be *simple* if no more than two curves intersect at any point.
Do simple, symmetric Venn diagrams with $n$ curves exist for every prime number $n$? | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Question 1
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,795 | AMR-142-0002 | Branko Grunbaum in Many Dimensions — Conjecture 2 | The Four Color Theorem for planar graphs is equivalent to the statement that the dual graph of every triangulated $2$-sphere is $3$-edge colorable. This is essentially Tait's 1880 observation that the Four Color Theorem is equivalent to showing that every cubic bridgeless planar graph is $3$-edge colorable [Tait1880].
... | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Conjecture 2
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,796 | AMR-142-0003 | Branko Grunbaum in Many Dimensions — Conjecture 3 | Still, this leaves the question open for genus $g$ with $1 \le g \le 4$. In particular, the following seems to be open.
If $ G $ is a simple loopless triangulation of a torus, then the dual graph of $ G $ is 3-edge-colorable. | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Conjecture 3
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,797 | AMR-142-0004 | Branko Grunbaum in Many Dimensions — Question 4 | A $2$-dimensional simplicial complex is said to be *polyhedral* if it admits an embedding in $\mathbb{R}^3$ with every vertex corresponding to a point, every edge corresponding to a straight line segment, and every triangle corresponding to a flat triangle contained in an affine plane. (It must also actually be a topol... | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Question 4
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,798 | AMR-142-0005 | Branko Grunbaum in Many Dimensions — Question 5 | Can every object in $\mathbb{R}^4$ be partitioned into $16$ equal parts with $4$ hyperplanes? In particular, can every set of $16n$ points in general position in $\mathbb{R}^4$ be separated into $16$ subsets of $n$ points, using 4 hyperplanes? | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Question 5
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,799 | AMR-142-0006 | Branko Grunbaum in Many Dimensions — Conjecture 6 (Ramos's conjecture) | According to Florian Frick, there is reason to think that the answer is yes. Let $\Delta(j,k)$ denote the minimum dimension $d$ such that every set of $j$ objects in $\mathbb{R}^d$ can be simultaneously partitioned into $2^k$ equal pieces by $k$ hyperplanes.
For every $j, k \ge 1$, $$\Delta(j,k) = \left\lceil \frac{2^... | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Conjecture 6
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Ramos's conjecture | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,800 | AMR-142-0007 | Branko Grunbaum in Many Dimensions — Conjecture 7 | [Gr\"unbaum--Kalai--Sarkaria] For every $d \ge 1$ there is a constant $C_d$ such that $$f_d (S)\le C_d f_{d-1}(S)$$ for every $d$-complex $S$ which is embeddable in $\mathbb{R}^{2d}$. | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Conjecture 7
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,801 | AMR-142-0008 | Branko Grunbaum in Many Dimensions — Conjecture 8 | Various formulations of this conjecture have apparently been discussed for some time---see, for example, Dey [Dey93]. Work of Kalai and Sarkaria [Kalai91,Sarkaria92] suggests a precise formulation, which would in particular give the best possible constant $C_d$ in every dimension. Gr\"unbaum wrote that this question is... | Source list: Kahle - Branko Grunbaum in Many Dimensions: Open Problems (2019)
Source item: Conjecture 8
Source URL: https://arxiv.org/abs/1901.08622
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Kahle | 2,019 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,802 | AMR-143-0001 | Geometric Hydrodynamics in Open Problems — Problem 1 | Find a variational (preferably geodesic) formulation describing the motion of an ideal fluid
in a fixed domain $M$ containing sources and sinks.
What is the correct formulation of the variational problem:
should one take into account the exterior forces and/or the ``memory" of the fluid? | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 1
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,803 | AMR-143-0002 | Geometric Hydrodynamics in Open Problems — Problem 2 | It turned out that in all of the above cases the corresponding configuration space has a natural groupoid structure.
Using the corresponding Lie groupoids of multiphase diffeomorphisms instead of the Lie group of
volume-preserving transformations in Arnold's setting one can describe the corresponding Lie algebroids and... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 2
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,804 | AMR-143-0003 | Geometric Hydrodynamics in Open Problems — Problem 3 | Describe the differential geometry (including computations of sectional and Ricci curvatures, conjugate points, etc.)
for the right-invariant (source-wise) $L^2$-metric on the Lie groupoid of multiphase or generalized volume-preserving diffeomorphisms (analogous to Arnold's description of the differential geometry of t... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 3
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,805 | AMR-143-0004 | Geometric Hydrodynamics in Open Problems — Problem 4 | Is it possible to extend Arnold's geodesic framework
from the group of diffeomorphisms $\mathscr{D}(M)$ to the semigroup of maps ${\mathcal{M}ap}(M)$
that would capture both smooth solutions and their continuations beyond emergence of shock waves
for Burgers-like and compressible fluid equations? | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 4
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,806 | AMR-143-0005 | Geometric Hydrodynamics in Open Problems — Problem 5 | Find a variational description of the system of material particles, moving in $\mathbb{R}^d$ with $ d>1$ and sticking upon collision,
for both potential and non-potential fluid flows.
This formulation should be sufficiently flexible to describe the formation and development of shock waves
in the system described by the... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 5
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,807 | AMR-143-0006 | Geometric Hydrodynamics in Open Problems — Problem 6 | In all other cases the metric is neither left- nor right-invariant. It would be interesting to investigate this system
for generic $S$ and $\rho(\omega)$. It would also be of interest to define and study a model system
where the configuration space is $SL(2)$, since this group looks more ``liquid-like".
A similar quest... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 6
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,808 | AMR-143-0007 | Geometric Hydrodynamics in Open Problems — Problem 7 (Two-point boundary value problem) | Let $M$ be a compact two-dimensional Riemannian manifold
and let $\varphi$ be a volume-preserving diffeomorphism of $M$ of Sobolev class $H^s$.
(i) \mathrm{(Surjectivity Problem)}
Find a divergence-free vector field $v \in H^s(M)$ such that $\exp_e{v} = \varphi$.
(ii) \mathrm{(Variational Problem)}
Find a curve $\gam... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 7
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Two-point boundary value problem | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,809 | AMR-143-0008 | Geometric Hydrodynamics in Open Problems — Problem 8 | Another interesting question is related to closed geodesics.
Suppose that $M$ is a compact surface (possibly with boundary) of genus at least $2$
or that $M$ is a multi-connected bounded domain in $\mathbb{R}^2$ with at least two holes.
Does there exist a closed geodesic in $\mathscr{D}^s_\mu(M)$? | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 8
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,810 | AMR-143-0009 | Geometric Hydrodynamics in Open Problems — Problem 9 | How ``far" is $\mathscr{D}_\mu(M)$ from being an infinite-dimensional (locally) symmetric Riemannian space? | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 9
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,811 | AMR-143-0010 | Geometric Hydrodynamics in Open Problems — Problem 10 | Are particle trajectories analytic in time in any dimension?
For fluids on manifolds with boundary how does this analyticity depends on whether the boundary is analytic or not? | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 10
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,812 | AMR-143-0011 | Geometric Hydrodynamics in Open Problems — Problem 11 | Prove that for any two-dimensional compact analytic Riemannian manifold $M$ with analytic boundary
the set $\mathscr{GM}(M)$ of generalized minimal flows is a nonempty and proper subset of
the Yudovich space $YU(M)$. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 11
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,813 | AMR-143-0012 | Geometric Hydrodynamics in Open Problems — Problem 12 | Prove that for any GM-flow $u(x,t)$ the level lines of vorticity $\mathrm{curl} u(x,t)=\text{const}$ are real-analytic. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 12
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,814 | AMR-143-0013 | Geometric Hydrodynamics in Open Problems — Problem 13 | Define anisotropic spaces more appropriate for hydrodynamics with larger smoothness in the direction of flow lines.
Prove existence and uniqueness theorems for ideal hydrodynamics in those function spaces in any dimension. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 13
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,815 | AMR-143-0014 | Geometric Hydrodynamics in Open Problems — Problem 14 | The concept of GM-flows appears closely related to the ``ancient flows" for the Navier--Stokes equations
but, in fact, it is different. Both concepts express the general idea that, in reality, we observe flows that have existed
since long time ago (one may imagine a river flow). In this regard, it is not natural to con... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 14
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,816 | AMR-143-0015 | Geometric Hydrodynamics in Open Problems — Problem 15 | Thus the structure of singularities of the exponential map in 2D hydrodynamics
resembles that of a smooth map between finite-dimensional manifolds.
This leads to the following natural question.
Quantify the failure of the Fredholm property of the $L^2$ exponential map in $\mathscr{D}^s_\mu(M)$
for a compact Riemannian... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 15
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,817 | AMR-143-0016 | Geometric Hydrodynamics in Open Problems — Problem 16 | - []
(a) Determine the order of the first conjugate point along any $L^2$ geodesic starting from the identity
in $\mathscr{D}_\mu^s(M)$.
(b) Is there a relation between existence of conjugate points in $\mathscr{D}_\mu^s(M)$
and Arnold stability criterion for stationary flows in $M$? | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 16
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,818 | AMR-143-0017 | Geometric Hydrodynamics in Open Problems — Problem 17 | Show that for any geodesic $\gamma(t)$ in $\mathscr{D}_\mu^s(M)$ its absolute complexity
is exponentially bounded above:
$$
\mathcal{C}(\gamma(t)) \lesssim e^{t\|\dot{\gamma}_0\|_{H^s}} .
$$ | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 17
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,819 | AMR-143-0018 | Geometric Hydrodynamics in Open Problems — Problem 18 (Aging Problem) | Consider an arbitrary solution $u=u(t,x)$ of the Euler equations in $M$.
Given any two time instants can one decide which velocity field $u(t_1,x)$ or $u(t_2,x)$
corresponds to an earlier time instant?
In other words, is it possible to determine the aging of the fluid from its velocity field?
Numerical experiments sugg... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 18
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Aging Problem | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,820 | AMR-143-0019 | Geometric Hydrodynamics in Open Problems — Problem 19 | Consider an ideal fluid in a periodic channel $M=\mathbb{T} \times (0,a)$.
Assume that its initial velocity $u_0$ is $C^1$ close to that of a plane-parallel flow
whose velocity profile $v=v(x_2)$ satisfies
$v'>0$ and $v''>0$ in $(0,a)$.
Suppose that the level lines of $\omega_0 = \mathrm{curl} u_0$ satisfy $\nabla\omeg... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 19
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,821 | AMR-143-0020 | Geometric Hydrodynamics in Open Problems — Problem 20 | Investigate properties of the entropy functions $H_\epsilon, H_{\epsilon,\delta}, \tilde{H}_\epsilon$
and $\tilde{H}_{\epsilon,\delta}$ in this section and explain relations between them. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 20
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,822 | AMR-143-0021 | Geometric Hydrodynamics in Open Problems — Problem 21 | In the two-dimensional case, besides the kinetic energy, this system has infinitely many enstrophy invariants.
These invariants are Casimir functionals: they do not depend on the metric on $M$, but specify a coadjoint orbit
(i.e. a particular set of isovortical vector fields) on which the Euler evolution takes place bu... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 21
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,823 | AMR-143-0022 | Geometric Hydrodynamics in Open Problems — Problem 22 | Justify the phenomenon observed in {\rm [ModViv19, ModViv21]} that integrable cases of point vortices
seem to be attractors for 2D Euler flows with generic smooth initial vorticity in 2D (for small numerical viscosity).
Design a model of dissipation in the 2D Euler equation which would produce such an integrable dynami... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 22
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,824 | AMR-143-0023 | Geometric Hydrodynamics in Open Problems — Problem 23 | Study in detail the motion and bifurcations of a small number of vortices on various manifolds:
half- and quarter-plane, hemi- and quarter-sphere, disk, torus, cylinder and half-cylinder, etc. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 23
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,825 | AMR-143-0024 | Geometric Hydrodynamics in Open Problems — Problem 24 | Study the first non-integrable cases for a small number of point vortices on non-orientable surfaces. Describe the full set of Casimirs and finite-dimensional approximations for the group of area-preserving diffeomorphisms of non-orientable surfaces, analogous to the ones in the orientable cases. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 24
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,826 | AMR-143-0025 | Geometric Hydrodynamics in Open Problems — Problem 25 | This is not the case for a general infinite-dimensional dynamical system
and, in particular, for the Eulerian hydrodynamics.
Nadirashvili showed that the Euler equation of a 2D fluid has wandering solutions:
there is an initial condition of a fluid in a 2D annulus whose neighbourhood never returns sufficiently closely
... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 25
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,827 | AMR-143-0026 | Geometric Hydrodynamics in Open Problems — Problem 26 | Relax the restrictions on the smoothness index $k$ of the $C^k$ spaces
(related to application of the KAM) and prove non-transitivity and non-mixing properties
of the 3D Euler in full generality. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 26
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,828 | AMR-143-0027 | Geometric Hydrodynamics in Open Problems — Problem 27 | Consider an integrable system on a compact $2n$-dimensional manifold, which
has $n$ first integrals in involution, functionally independent almost everywhere.
How wild could such a system be on a singular submanifold, where the integrals become dependent? What are constraints dictated by the integrability? Could one ob... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 27
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,829 | AMR-143-0028 | Geometric Hydrodynamics in Open Problems — Problem 28 | Arnold pointed out the remarkable topology of steady 3D fields:
for an analytic steady field, not everywhere collinear with its curl, the flow domain is almost everywhere
fibered into invariant tori and annuli, see [arn66, ak]. If the steady field is everywhere collinear with its vorticity
but the proportionality coeff... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 28
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,830 | AMR-143-0029 | Geometric Hydrodynamics in Open Problems — Problem 29 ([ak]) | Find a sequence of higher helicity invariants for vector fields so that, given a field, if all the previous invariants are equal to zero for it, then the first nonzero invariant bounds the field's energy from below. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 29
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: [ak] | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,831 | AMR-143-0030 | Geometric Hydrodynamics in Open Problems — Problem 30 | Find a direct link from 3D NLS equation to 1D NLS equation as a reduction to singular solutions,
rather than going through the LIA procedure.
A similar question arises for the compressible Euler equations as a reduction
from 3D to singular solutions supported on 1D submanifolds. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 30
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,832 | AMR-143-0031 | Geometric Hydrodynamics in Open Problems — Problem 31 | Geometric description of vortex sheets leads to an interesting non-local metric of hydrodynamical pedigree
on shape spaces:
it is an $H^{-1/2}$-metric on closed hypersurfaces bounding the same volume
(or equivalently, on constant densities inside those hypersurfaces),
see [IK18]. Such a metric is constructed with the h... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 31
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,833 | AMR-143-0032 | Geometric Hydrodynamics in Open Problems — Problem 32 | Extend the above results on symplectomorphism and the K\"ahler map to the wave functions with zeros on $M$.
Explain the quantization condition controversy {\rm [Wallstrom, Fritsche]} in the language of the momentum map
for the above semi-direct product group. | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 32
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,834 | AMR-143-0033 | Geometric Hydrodynamics in Open Problems — Problem 33 | The connection between equations of quantum mechanics
and hydrodynamics might shed some light on the hydrodynamic quantum analogues studied,
e.g., in [Couder, Bush]:
the motion of bouncing droplets in certain vibrating liquids manifests many properties of quantum mechanical particles.
Explain the droplet-quantum parti... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 33
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,835 | AMR-143-0034 | Geometric Hydrodynamics in Open Problems — Problem 34 | Now, the next natural question concerns what happens with the energy that has escaped from the system?
One can introduce the ``limit absorption principle'' for our system. Namely, one can introduce a friction for the balls
of the family $F_N$ which absorbs the energy and then let $N$ go to infinity. But this solution i... | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 34
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,836 | AMR-143-0035 | Geometric Hydrodynamics in Open Problems — Problem 35 | Is there a similar mechanism which describes a cascade in real hydrodynamics?
Is it possible to define the parameters $n_i, m_i,$ and $ r_i$ in such a way that the above apparatus works as intended? | Source list: Khesin, Misiolek, Modin - Geometric Hydrodynamics in Open Problems (2022)
Source item: Problem 35
Source URL: https://arxiv.org/abs/2205.01143
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Khesin, Misiolek, Modin | 2,022 | 9 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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"
} | {
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,837 | AMR-144-0001 | Randomstrasse101 Open Problems (2024) — Conjecture 2 (Globally Synchronizing Regular Graphs) | Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend [ABKSST-SynchExpander] recently showed that a random Erd\mathbb{H}{o}s-R\'enyi graph is globally synchronizing above the connectivity threshold with high probability (you can also see the Quanta article covering this at https://www.quantamagazine.org/new-proof-... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 2
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Globally Synchronizing Regular Graphs | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,838 | AMR-144-0002 | Randomstrasse101 Open Problems (2024) — Conjecture 3 (Global Synchrony with negative edges) | Given any $\varepsilon>0$, the $n\times n$ random matrix $A$ with zero diagonal and off-diagonal entries given by
\[
A_{ij} =
\left\{ \begin{array}{ccl}
1 & \text{with probability} & \frac12+\delta\\
-1 & \text{with probability} &
\frac12-\delta,
\end{array} \right.
\]
with $\delta \geq (1+\varepsilon)\sqrt{\frac{\log ... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 3
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Global Synchrony with negative edges | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,839 | AMR-144-0003 | Randomstrasse101 Open Problems (2024) — Conjecture 4 (Density threshold for Global Synchrony) | Another elusive question about global synchrony is the ``extremal combinatorics'' version of the question.
For any $\varepsilon>0$, there exists $n>0$ and a graph $G$ on $n$ nodes such that the minimum degree of $G$ is at least $\left(\frac34-\varepsilon\right)n$ and $G$ is not globally synchronizing. | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 4
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Density threshold for Global Synchrony | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,840 | AMR-144-0004 | Randomstrasse101 Open Problems (2024) — Conjecture 5 (Ellipsoid fitting) | Let $n, d \geq 1$ and $x_1, \cdots, x_n \sim \mathcal{N}(0,\mathrm{I}_d/d)$. For any $\varepsilon > 0$:
\begin{align}
\limsup_{d \to \infty} \frac{n}{d^2} \leq \frac{1-\varepsilon}{4} \Rightarrow \lim_{d \to \infty} \mathbb{P}[\exists \mathcal{E} \textrm{ an ellipsoid fit to } (x_i)_{i=1}^n] = 1, \\
\liminf_{d \to \i... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 5
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Ellipsoid fitting | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,841 | AMR-144-0005 | Randomstrasse101 Open Problems (2024) — Problem 6 | Consider synchronization model over $SO(2)$ or synchronization model over a finite group, where the signal $x$ is sampled uniformly over group elements. Find a scaling $L= L_n$ of number of frequencies such that there exists a polynomial-time algorithm that can detect the signal reliably for all $\lambda>\lambda_{\text... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Problem 6
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,842 | AMR-144-0006 | Randomstrasse101 Open Problems (2024) — Conjecture 8 (Kikuchi Spectral Threshold) | Given $r,\ell, n$ positive integers satisfying $n\gg \ell\geq \frac{r}2$ and r even ($r$ and $\ell$ will be fixed and $n\to\infty$). For each $S\subset [n]$ with $|S|=r$ let $Z_S\sim\mathcal{N}(0,1)$, and all independent. Let $\lambda\geq 0$ and let $M$ be the $\binom{n}{\ell} \times \binom{n}{\ell}$ matrix (with rows ... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 8
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Kikuchi Spectral Threshold | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,843 | AMR-144-0007 | Randomstrasse101 Open Problems (2024) — Conjecture 9 (K\'{o}mlos Conjecture) | When the condition of $\pm1$ entries is replaced by an $\ell_2$ condition on the columns, the corresponding question is the famous K\'{o}mlos Conjecture (which coincidentally is the first open problem in my lecture notes from around a decade ago [Afonso10L42P]).
There exists a universal constant $K$ such that, for all... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 9
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: K\'{o}mlos Conjecture | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,844 | AMR-144-0008 | Randomstrasse101 Open Problems (2024) — Problem 10 (How many deviations are needed afterall?) | It is easy to see that the statement of this conjecture implies Spencer's Theorem, since the columns of an $n\times n$, $\pm1$ matrix, have $\ell_2$ norm $\sqrt{n}$. The best current lower bound for K\'{o}mlos is a recent construction by Tim Kunisky [Tim-lowerboundKomlos] which shows a lower bound on $K$ of $1+\sqrt{2}... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Problem 10
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: How many deviations are needed afterall? | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,845 | AMR-144-0009 | Randomstrasse101 Open Problems (2024) — Conjecture 11 | Also, to the best of my knowledge, there is no known construction (or proof of existence) of an infinite family achieving a lower bound strictly larger than one. More explicitly, we formulate the following conjecture.
Given the definitions above, we have
$$\limsup_{n\to\infty}\sup_{A\in\{\pm1\}^{n\times n}}\frac1{\sqr... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 11
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,846 | AMR-144-0010 | Randomstrasse101 Open Problems (2024) — Conjecture 12 | Let $H_k$ be the $2^k\times 2^k$ Hadamard matrix obtained by the Sylvester construction (see above). For odd $k$, we have
\[
\mathrm{disc}(H_k) = \sqrt{2}\sqrt{2^k}.
\] | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Conjecture 12
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,847 | AMR-144-0011 | Randomstrasse101 Open Problems (2024) — Problem 14 | Consider the SK-model with an arbitrary external field. Does Glauber Dynamics fast mix (meaning in polynomial time) up to the replica symmetry breaking threshold $\beta_\ast =1$? | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2024 (2025)
Source item: Problem 14
Source URL: https://arxiv.org/abs/2504.20539
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,025 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,848 | AMR-145-0001 | Randomstrasse101 Open Problems (2025) — Conjecture 1 (Sharp graph matrix bounds) | However, there are still examples of shapes $\alpha$ for which iterated linear inequalities fail to produce sharp bounds. Moreover, there are examples of shapes whose norm bounds do have additional non-constant factors. We conjecture that given a fixed shape $\alpha$, the norm of the associated graph matrix grows as a ... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 1
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Sharp graph matrix bounds | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,849 | AMR-145-0002 | Randomstrasse101 Open Problems (2025) — Conjecture 1 | An interesting question is to characterize the Lov\'asz number for random graphs.
A well-known family of random graphs is the Erd\mathbb{H}{o}s-R\'enyi random graphs \(G(n, p)\). Here, \(n\) denotes the number of vertices, and \(0 \leq p \leq 1\) is the probability parameter. To sample a graph from \(G(n, p)\), for eac... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 1
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,850 | AMR-145-0003 | Randomstrasse101 Open Problems (2025) — Conjecture 2 | We conjecture, based on numerical observations, that the Lov\'asz number for random dense circulant graphs behaves similarly to that for \(G(n, 1/2)\). This motivates the following question.
Let \(G\) be distributed as a random dense circulant graph. Then,
\begin{equation}
\mathbb{E} \vartheta(G) = (1 + o(1))\sqrt{n}.... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 2
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,851 | AMR-145-0004 | Randomstrasse101 Open Problems (2025) — Conjecture 1 | In [Bandyopadhyay2002-MUBs] it is shown that $\mathrm{MUB}(d)\leq d+1$ and that equality is achieved when $d$ is a prime power. For $d$ not a prime power, the problem of determining $\mathrm{MUB}(d)$ is still open. The smallest open instance is particularly well known for being a tantalizing open problem that has remai... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 1
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,852 | AMR-145-0005 | Randomstrasse101 Open Problems (2025) — Problem 2 | An interesting approach to this problem is to try to build Sum-of-Squares proofs of $\mathrm{MUB}(6)<7$. In [BDK-2022-DualBoundsMUB] it was shown that a certain relaxation related to degree-2 Sum-of-Squares cannot prove that $M(6) < 7$. It is however unclear whether higher degree levels of Sum-of-Squares can provide su... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Problem 2
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,853 | AMR-145-0006 | Randomstrasse101 Open Problems (2025) — Conjecture 1 | Let $\omega(G_p)$ denote the clique number of $G_p$ i.e. the size of the largest set of nodes that are all pairwise connected by an edge.
Since $G_p$ is isomorphic to its complement this is the same as $\alpha(G_p)$ the independence number of $G_p$ (the size of the largest set of nodes that have no edges between them).... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 1
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,854 | AMR-145-0007 | Randomstrasse101 Open Problems (2025) — Conjecture 2 | We start with the notion of localization [MMP-2019-LPCliquesPaley,Kunisky-2023-SpectralPseudorandomnessPaleyClique]: Let $G_{p,1}$ denote the induced subgraph on the vertices adjacent to the node 0, which we call the 1-localization. Because the 1-localization does not depend on the node picked (see [Kunisky-2023-Spectr... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 2
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,855 | AMR-145-0008 | Randomstrasse101 Open Problems (2025) — Conjecture 3 | For $p \equiv 1 \pmod{4}$ prime and large enough we have $\vartheta(\overline{G_{p,2}}) \leq \frac23 \sqrt{p}$ where $G_{p,2}$ is the 2-localization of the Paley Graph described above. | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 3
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,856 | AMR-145-0009 | Randomstrasse101 Open Problems (2025) — Conjecture 4 | There is another approach to to use convex relaxation to potentially improve over upper bounds on the $
\omega(G_p)$. Recall that the Lovasz $\vartheta$ function can be viewed a as degree-2 Sum-of-Squares relaxation of the clique (or independence) number. A fascinating question is whether the degree-4 Sum-of-Squares re... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 4
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,857 | AMR-145-0010 | Randomstrasse101 Open Problems (2025) — Conjecture 5 | The Paley Equiangular Tight Frame construction satisfies the Restricted Isometry Property beyond the square-root bottleneck. In other words, there exists $\varepsilon>0$ such that, for $m \sim p^{1/2+\varepsilon}$, any matrix formed by picking $m$ (distinct) vectors in the Paley ETF has condition number bounded by a co... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 5
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,858 | AMR-145-0011 | Randomstrasse101 Open Problems (2025) — Conjecture 1 | Let $N=4M-5$ and draw $A\in\mathbb{C}^{N\times M}$ at random (here the important thing is for $ \mathrm{im}(A) $ to be drawn uniformly from the Grassmannian of $ M $-dimensional subspaces of $ \mathbb{C}^{4M-5}$, it can e.g. be with iid gaussian entries). Let $p_M$ be the probability that the mapping $ x \bmod \mathbb{... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 1
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,859 | AMR-145-0012 | Randomstrasse101 Open Problems (2025) — Conjecture 2 | \begin{definition}
Given $A\in\mathbb{R}^{N\times M}$ let $\omega(A)$ be defined as
\begin{equation}
\omega(A) = \min_{S\subseteq [N]: \mathrm{rank}(A_{S^c})<n} \sigma_M(A_S),
\end{equation}
where $A_S$ denotes the submatrix made of the rows indexed by $S$ and $\sigma_M$ is the $M$-th singular value (which is the small... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 2
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,860 | AMR-145-0013 | Randomstrasse101 Open Problems (2025) — Problem 3 | This also motivates the following question (for which I do not know the state of the art, but a quick search did not reveal that the answer is already known).
What is the value of $\omega(A)$ for $A$ with iid standard gaussian entries? Does it satisfy the bound in the conjecture? If so, what is $\beta$? | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Problem 3
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,861 | AMR-145-0014 | Randomstrasse101 Open Problems (2025) — Conjecture 1 (Type-$2$ constant of Tensors) | Let $p \geq 2$, then
\begin{equation}
\mathbb{E} \left| \left| \sum_{i=1}^n g_i T_i \right| \right|_{ \mathcal{I}_p} \leq \tilde{\mathcal{O}}_{r,p} \left( d^{\frac12 - \frac1p } \sqrt{\sum_{i=1}^n \left| \left| T_i \right| \right|_{ \mathcal{I}_p}^2 }\right).
\end{equation}
(Here $\tilde{\mathcal{O}}_{r,p}$ hides multi... | Source list: Bandeira et al. - Randomstrasse101: Open Problems of 2025 (2026)
Source item: Conjecture 1
Source URL: https://arxiv.org/abs/2603.29571
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Type-$2$ constant of Tensors | 3 | open | Bandeira et al. | 2,026 | 19 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | 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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,862 | AMR-146-0001 | Open Problems in Algebraic Statistics — Specific Problem 1 | Our first question concerns three-dimensional contingency tables
$(p_{ijk})$ whose indices $i,j,k$ range over a set of four elements,
such as the set $ \{{\tt A},{\tt C},{\tt G},{\tt T}\}$ of DNA bases.
}
{\em Consider the
variety of $4 {\times} 4 {\times} 4$-tables
of tensor rank at most $4$.
There are certain known ... | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: Specific Problem (#1)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,863 | AMR-146-0002 | Open Problems in Algebraic Statistics — General Problem 2 | This particular open problem appears in [ASCB, Conjecture 3.24],
and it here serves as a placeholder for the following broader direction of inquiry.
*Study the geometry and commutative algebra
of graphical models with hidden random variables.
Construct these varieties by gluing familiar
secant varieties, and by applyi... | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: General Problem (#2)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,864 | AMR-146-0003 | Open Problems in Algebraic Statistics — Specific Problem 3 | } *Find a geometric characterization of those projective varieties
whose maximum likelihood degree (ML degree) is equal to one.*
This question and others raised in [CHKS, HKS]
are just the tip of an iceberg: | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: Specific Problem (#3)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,865 | AMR-146-0004 | Open Problems in Algebraic Statistics — General Problem 4 | *Study the geometry of maximum likelihood estimation
for algebraic statistical models. * | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: General Problem (#4)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,866 | AMR-146-0005 | Open Problems in Algebraic Statistics — Specific Problem 5 | {\em
Which sets of almost-principal minors can be zero
for a positive definite symmetric $5 {\times} 5$-matrix?} | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: Specific Problem (#5)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,867 | AMR-146-0006 | Open Problems in Algebraic Statistics — General Problem 6 | *Study the geometry of conditional independence models
for multivariate Gaussian random variables.* | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: General Problem (#6)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,868 | AMR-146-0007 | Open Problems in Algebraic Statistics — General Problem 7 | } {\em Characterize the image of the entropy map $ H $
and how it intersects the various faces of ${\tt SubMod}_n$.
Study the fibers of this map.} | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: General Problem (#7)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,869 | AMR-146-0008 | Open Problems in Algebraic Statistics — Bonus Problem 8 | } {\em Does every strict CI model have a $\mathbb{Q}$-rational point?} | Source list: Sturmfels - Open Problems in Algebraic Statistics (2007)
Source item: Bonus Problem (#8)
Source URL: https://arxiv.org/abs/0707.4558
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Sturmfels | 2,007 | 5 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"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": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.