Daily Papers

We prove that the homotopy limits and homotopy colimits of chain complexes can be computed by the cobar and bar constructions. We also show that the totalizations of double complexes compute the homotopy limits and homotopy colimits of simplicial and cosimplicial chain complexes.

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis alpha-entropy, depending on the value of the parameter.

Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network-based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve silver-medal level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary constructions in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving gold-medal level performance and surpassing AlphaGeometry, a competitive neural network-based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo-409, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.

We present a statistical analysis of the connections between galactic bars, star formation, and active galactic nucleus (AGN) activity using 33,201 disk galaxies (0.01 < z < 0.05) from DESI DR1 cross-matched with Galaxy Zoo DESI. Based on morphological classifications, we identify 3,508 strongly barred and 8,335 weakly barred systems. We find that barred galaxies exhibit a clear bimodal distribution in color-mass space: weak bars are preferentially found in bluer, lower-mass disks, whereas strong bars are more common in massive, redder systems. Strongly barred galaxies are on average more massive and metal-rich than unbarred systems. In addition, strong bars enhance central SFRs in low-mass galaxies but reduce sSFRs in massive systems, reflecting a dual role where bars initially trigger central star formation but eventually promote quenching by accelerating gas consumption. In terms of nuclear activity, barred galaxies display a higher incidence of AGN activity. The presence of a bar is also associated with an increased fraction of powerful AGN, with the highest proportions found in strongly barred systems. However, the correlations between AGN activity and detailed bar structural parameters are weak, suggesting that the link between bars and nuclear activity is indirect and regulated by multiple factors. Overall, our results support a scenario in which bars facilitate angular-momentum transport and gas inflow, thereby driving central star formation and fueling supermassive black hole accretion while operating alongside other processes that shape galaxy evolution.

The Milky Way is known to contain a stellar bar, as are a significant fraction of disc galaxies across the universe. Our understanding of bar evolution, both theoretically and through analysis of simulations indicates that bars both grow in amplitude and slow down over time through interaction and angular momentum exchange with the galaxy's dark matter halo. Understanding the physical mechanisms underlying this coupling requires modelling of the structural deformations to the potential that are mutually induced between components. In this work we use Basis Function Expansion (BFE) in combination with multichannel Singular Spectral Analysis (mSSA) as a non-parametric analysis tool to illustrate the coupling between the bar and the dark halo in a single high-resolution isolated barred disc galaxy simulation. We demonstrate the power of mSSA to extract and quantify explicitly coupled dynamical modes, determining growth rates, pattern speeds and phase lags for different stages of evolution of the stellar bar and the dark matter response. BFE & mSSA together grant us the ability to explore the importance and physical mechanisms of bar-halo coupling, and other dynamically coupled structures across a wide range of dynamical environments.