We show that the Itô solutions of the nonlinear stochastic heat equation $$ \partial_t u^\varepsilon- Δu^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla ξ_\varepsilon, $$ where $ ξ_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $ξ$, converge in law, as $\varepsilon\to 0$, to the solution of the stochastic heat equation with right-hand side $cg'g(u)ξ$ with a constant $c>0$. Since the noise $\nablaξ$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques.

This paper is devoted to the variational derivation of reduced models for elastic membranes with fracture under constraints on the determinant of the deformation gradient. We consider two physically relevant settings: the non-interpenetration regime, in which the deformation is required to be orientation-preserving ($\det \nabla u > 0$), and the incompressible regime, in which the deformation preserves volume ($\det \nabla u = 1$). In both cases, the surface energy density is allowed to depend on the jump amplitude, thus encompassing cohesive fracture models with activation threshold. The main technical contribution is the construction of recovery sequences that simultaneously satisfy the determinant constraint and optimize the surface energy. This is achieved through a combination of $C^\infty$ diffeomorphisms converging to the identity (which rotate the normal to the jump set so as to minimize the reduced surface energy), and a new smooth approximation result for $GSBV^p$ functions.

We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge. This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam$(F(P))$ for sets $P$ of $n$ points in convex position. The current best bounds are $\frac{14}{9}n-O(1) \leq$ diam$(F(P))<\frac{15}{9}n-3$ [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called *conflict graphs*, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms). In this paper, we pick up the concept of conflict graphs and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint). Extending the line of research from [BKUV SODA25], we present new insights on the diameter of the flip graph. Their lower bound is based on a constant-size pair of trees, one of which is *stacked*. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most $\frac{14}{9}(n-1)$ to any other tree. Lastly, we improve the lower bound on the diameter of the flip graph $F(P)$ for $n$ points in convex position to $\frac{11}{7}n-o(n)$.

We introduce and study the notion of entropy of affine permutations and prove that it coincides with the atomic length associated with the sum of the fundamental weights for a type $A$ affine root system, as defined by the first two authors. We then establish an analogue of the Granville-Ono theorem by showing that any nonnegative integer can be realised as the entropy of an affine permutation or alternatively, as the size of a core multipartition as introduced by the last two authors. Our proof uses an additive combinatorics theorem due to Hall on difference sets of permutations modulo $n$. More generally, we give a polynomial expression of the atomic length associated with any dominant weight in affine type $A$ and investigate the problem of its universality. Beyond type $A$, we are able to prove that the entropy of affine type $C_n$ permutations is universal when $2n+1$ is prime. This is achieved by establishing an analogue of Hall's theorem for the hyperoctahedral group based on Alon's combinatorial Nullstellensatz. We also propose conjectures generalising the results presented in the paper, each supported by computational evidence. Finally, we show that in any affine classical type, the problem of the universality of the atomic length simplifies in large rank when the weight considered is conveniently adjusted.

In this paper, we explicitly classify the corank 4 unitary representations of symplectic or split odd special orthogonal groups over non-Archimedean local fields of characteristic zero, by classifying Arthur representations of corank 4 and verifying the corresponding unitary dual conjecture recently proposed by Hazeltine-Jiang-Liu-Lo-Zhang in [HJLLZ24].

We present a new approach to hyperbolic plugs, via a construction of bicontact plugs on 3-manifolds with boundary that are surface bundles over the circle. The boundary components are quasi transverse tori, and we prove a gluing theorem that allows us to produce closed manifolds carrying new transitive Anosov flows. We show that a toroidal manifold produced by gluing two copies of the figure eight knot complement may carry many nonequivalent Anosov flows, and likewise a manifold composed of a figure eight complement and a trefoil complement. We further show that certain generalized Handel--Thurston surgeries can be realized as sequences of Goodman--Fried surgeries and produce new examples of different surgery sequences resulting in the same Anosov flow.

Diffusion language models (DLMs) have emerged as a promising alternative to autoregressive (AR) models for language modeling, allowing flexible generation order and parallel generation of multiple tokens. However, this flexibility introduces a challenge absent in AR models: the \emph{decoding strategy} -- which determines the order and number of tokens generated at each iteration -- critically affects sampling efficiency. Among decoding strategies explored in practice, confidence-based methods, which adaptively select which and how many tokens to unmask based on prediction confidence, have shown strong empirical performance. Despite this success, our theoretical understanding of confidence-based decoding remains limited. In this work, we develop the first theoretical analysis framework for confidence-based decoding in DLMs. We focus on an entropy sum-based strategy that continues unmasking tokens within each iteration until the cumulative entropy exceeds a threshold, and show that it achieves $\varepsilon$-accurate sampling in KL divergence with an expected number of iterations $\widetilde O(H(X_0)/\varepsilon)$, where $H(X_0)$ denotes the entropy of the target data distribution. Notably, this strategy yields substantial sampling acceleration when the data distribution has low entropy relative to the sequence length, while automatically adapting to the intrinsic complexity of data without requiring prior knowledge or hyperparameter tuning. Overall, our results provide a theoretical foundation for confidence-based decoding and may inform the design of more efficient decoding strategies for DLMs.

Many natural and social science systems are described using probability distributions over elements that are related to each other: for instance, occupations with shared skills or species with similar traits. Standard information theory quantities such as entropies and $f$-divergences treat elements interchangeably and are blind to the similarity structure. We introduce a family of divergences that are sensitive to the geometry of the underlying domain. By virtue of being the Bregman divergences of structure-aware entropies, they provide a framework that retains several advantages of Kullback-Leibler divergence and Shannon entropy. Structure-aware divergences recover planted patterns in a synthetic clustering task that conventional divergences miss and are orders of magnitude faster than optimal transport distances. We demonstrate their applicability in economic geography and ecology, where structure plays an important role. Modelling different notions of occupation relatedness yields qualitatively different regionalisations of their geographic distribution. Our methods also reproduce established insights into functional $β$-diversity in ecology obtained with optimal transport methods.

We give a sufficient combinatorial condition for the non-negativity of the coefficients of polynomial quotients of products of $q$-integers, also known as cyclotomic generating functions (CGFs). This slightly extends work by Iano-Fletcher, Pizzato, Sano and Tasin, who studied this condition as a criterion for quasismoothness of complete intersections in weighted projective spaces. As a consequence, we solve a problem by Billey and Swanson, prove most cases of an unpublished conjecture by Stanton and most cases of two conjectures by Gatzweiler and Krattenthaler. We also study sufficient conditions given by structural properties of the division lattice.

We consider random polytopes in the $d$-dimensional Euclidean space that are the convex hulls i.i.d. random points selected according to beta-prime distributions. These distributions are rotationally symmetric, heavy-tailed, and their support is the entire space, making them distinct from other commonly studied distributions, for instance, the uniform and Gaussian distributions. We prove lower bounds for the variances of the intrinsic volumes and the $f$-vector of such random polytopes. Beta-prime random polytopes are the push-forwards of spherical random polytopes, which are the convex hulls of random points chosen in the upper open hemisphere according to some rotationally symmetric distribution, including the uniform distribution in the open half-sphere. Our variance lower bounds also transfer to the spherical settings.

The Takagi function $T:[0,1]\to \mathbb{R}$ is a classical example of a continuous nowhere differentiable function. In this paper, we study the discrete dynamical system generated by the Takagi function. First, we prove that for almost every point $x\in [0,1]$, the orbit $(T^n(x))_n$ converges to $2/3$. We introduce the family of Takagi maps, given by $\textbf{T}_γ=γ\cdot T$, where $γ>0$ is a parameter. We also study the shadowing property for this family of maps. We show that the Takagi function has the shadowing property. Additionally, we provide two distinct techniques that allow us to find values of the parameter $γ$ for which $\textbf{T}_γ$ fails to have the shadowing property. Finally, we pose some open questions.

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced separator of size $O_d(m^{1/d}n^{1-2/d})$. This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.

By providing a suitable generalization of Newman's bijective correspondence known for cocommutative Hopf algebras, we prove that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, once faithful (co)flatness conditions are satisfied. This result unifies and generalizes the semi-abelianness of cocommutative Hopf algebras and of cocommutative color Hopf algebras known up to now. As a consequence of the semi-abelianness, the category of cocommutative Hopf monoids is also action representable. Finally, we prove that abelian objects in the category of cocommutative Hopf monoids coincide exactly with commutative and cocommutative Hopf monoids, which form so an abelian category.

This paper re-evaluates Jozsef Sutak (1865-1954), a Hungarian scholar-priest and professor, as a grey eminence rather than a genius, offering a counter-narrative to the history of Hungarian university mathematics. By examining his career - including his 1897 Bolyai translation and his defense of set theory during the 1911 Grundlagenkrise - the study illuminates the overlooked substructure of the academic system. Key institutional moments, such as his 1912 appointment over Frigyes Riesz and Alfred Haar and his administrative role during the Numerus Clausus era, reveal a system prioritizing rigorous pedagogy and stability over avant-garde research. Sutak's legacy is the foundation and ethical commitment that enabled the next generation of Hungarian mathematical giants to emerge.

This is the fourth paper in the CayleyPy project, which applies AI methods to the exploration of large graphs. In this work, we suggest the existence of a new discrete version of holographic string dualities for this setup, and discuss their relevance to AI systems and mathematics. Many modern AI tasks -- such as those addressed by GPT-style language models or RL systems -- can be viewed as direct analogues of predicting particle trajectories on graphs. We investigate this problem for a large family of Cayley graphs, for which we show that surprisingly it admits a dual description in terms of discrete strings. We hypothesize that such dualities may extend to a range of AI systems where they can lead to more efficient computational approaches. In particular, string holographic images of states are proposed as natural candidates for data embeddings, motivated by the "complexity = volume" principle in AdS/CFT. For Cayley graphs of the symmetric group S_n, our results indicate that the corresponding dual objects are flat, planar polygons. The diameter of the graph is equal to the number of integer points inside the polygon scaled by n. Vertices of the graph can be mapped holographically to paths inside the polygon, and the usual graph distances correspond to the area under the paths, thus directly realising the "complexity = volume" paradigm. We also find evidence for continuous CFTs and dual strings in the large n limit. We confirm this picture and other aspects of the duality in a large initial set of examples. We also present new datasets (obtained by a combination of ML and conventional tools) which should be instrumental in establishing the duality for more general cases.

We express the Kodaira-Iitaka dimension and the multiplicity of graded linear series in terms of the intersection theory of the plurisubharmonic envelope associated with the linear series, and obtain two refined versions of these formulas at the pointwise and at the metric levels. At the pointwise level, we focus on the weak convergence of the partial Bergman kernel associated with the linear series and a Bernstein-Markov measure. At the metric level, we compute the asymptotic ratio of the volumes of unit balls defined by the sup-norms on the linear series. Based on our findings, we introduce a non-pluripolar version of the numerical Kodaira-Iitaka dimension for a line bundle, show that this invariant dominates the classical Kodaira-Iitaka dimension and is, in turn, bounded above by the numerical versions proposed so far.

In this work, we show that for all statistical estimation problems, a natural MMSE instability (discontinuity) condition implies the failure of stable algorithms, serving as a version of OGP for estimation tasks. Using this criterion, we establish separations between stable and polynomial-time algorithms for the following MMSE-unstable tasks (i) Planted Shortest Path, where Dijkstra's algorithm succeeds, (ii) random Parity Codes, where Gaussian elimination succeeds, and (iii) Gaussian Subset Sum, where lattice-based methods succeed. For all three, we further show that all low-degree polynomials are stable, yielding separations against low-degree methods and a new method to bound the low-degree MMSE. In particular, our technique highlights that MMSE instability is a common feature for Shortest Path and the noiseless Parity Codes and Gaussian subset sum. Last, we highlight that our work places rigorous algorithmic footing on the long-standing physics belief that first-order phase transitions--which in this setting translates to MMSE-instability impose fundamental limits on classes of efficient algorithms.

Simulation methods have become important tools for quantifying partisan and racial bias in redistricting plans. We generalize the Sequential Monte Carlo (SMC) algorithm of McCartan and Imai (2023), one of the commonly used approaches. First, our generalized SMC (gSMC) algorithm can split off regions of arbitrary size, rather than a single district as in the original SMC framework, enabling the sampling of multi-member districts. Second, the gSMC algorithm can operate over various sampling spaces, providing additional computational flexibility. Third, we derive optimal-variance incremental weights and show how to compute them efficiently for each sampling space. Finally, we incorporate Markov chain Monte Carlo (MCMC) steps, creating a hybrid gSMC-MCMC algorithm that can be used for large-scale redistricting applications. We demonstrate the effectiveness of the proposed methodology through analyses of the Irish Parliament, which uses multi-member districts, and the Pennsylvania House of Representatives, which has more than 200 single-member districts.

Let $\mathbb{F}_\ell$ be a finite field with $\ell$ elements and let $G = C_p \rtimes C_m$ be a faithful split metacyclic group. In this paper, we develop a complete theory for the twisted group algebra $\mathbb{F}_\ell^αG$. Using the Lyndon--Hochschild--Serre spectral sequence, we prove that the second cohomology group of $G$ is isomorphic to $\mathbb{F}_\ell^\times/(\mathbb{F}_\ell^\times)^m$, and we show that all twisting occurs only on the $C_m$ factor. We determine the primitive central idempotents by analyzing the combined action of the Frobenius automorphism and the group action on the character group of $C_p$. Using crossed product theory and the structure of finite fields, we obtain the complete Wedderburn decomposition of $\mathbb{F}_\ell^αG$ into matrix algebras over explicitly determined fields $\mathbb{F}_{\ell^{d_j}}$. Finally, the irreducible projective representations of $G$ over $\mathbb{F}_\ell$ are also determined.

Cross-diffusion systems play a central role in mathematical modelling, in which density-dependent dispersal and multiscale mechanisms can lead to spatial segregation and diffusion-driven instabilities. In several relevant examples, including generalised SKT-type competition models, cross-diffusion terms can be rigorously derived as fast-reaction limits, thereby providing a clear biological interpretation while posing significant analytical challenges. In this work, we investigate the impact of biologically derived cross-diffusion on Turing instability. For a generalised SKT framework, we characterise instability conditions for a broad class of cross-diffusion functions arising from fast-reaction mechanisms. We then propose an alternative fast-reaction formulation leading to a different diffusion structure and show that, in this case, diffusion-driven pattern formation is prevented. We further discuss an example motivated by dietary diversity and starvation dynamics, and analyse how the sign structure of the reaction Jacobian interacts with cross-diffusion in determining the onset of patterns. Our results clarify structural features that promote or inhibit spatial self-organisation in competitive systems.

We consider a diffuse-interface model for two-phase incompressible viscous flows with a soluble surfactant in a bounded porous medium. This hydrodynamic system consists of a Darcy--Forchheimer equation for the seepage velocity $\boldsymbol{u}$ coupled with two Cahn--Hilliard equations involving Flory--Huggins type singular potentials, one for the phase-field variable $ϕ$, the difference in volume fractions of the two fluids, and the other for the surfactant concentration $ψ$. We study the initial boundary value problem in two or three dimensions, with impermeability boundary conditions for $\boldsymbol{u}$ and homogeneous Neumann boundary conditions for $(ϕ, ψ)$ and their associated chemical potentials. First, we establish the existence of global weak solutions via an implicit-explicit time-discretization scheme based on the energy dissipation law. Furthermore, applying the seminal results of the first and third authors (arXiv:2510.17296), we prove that every weak solution satisfying an energy inequality converges to a single equilibrium as time tends to infinity. In sharp contrast with the available literature on similar models, in this case weak solutions are enough to guarantee the uniqueness of asymptotic limits, without the necessity of any further eventual regularization.

Given a Nori motivic local system over a smooth, connected complex algebraic variety, we define its exceptional locus as a way to measure the variation in the motivic complexity of its stalks. The definition is given explicitly in terms of motivic Galois groups and Artin motives. Our main result is a motivic analogue of the Cattani--Deligne--Kaplan Theorem, asserting that the exceptional locus is a countable union of closed algebraic subvarieties. Moreover, we show that the maximal such subvarieties are defined over any algebraically closed subfield over which the ambient variety and the motivic local system admit models, and that they are stable under Galois conjugation when these models descend to a further subfield. This strengthens and extends previous results by André in the pure case. We obtain a similar geometric description for the splitting locus of the motivic weight filtration. In the case of 1-motivic local systems, the above properties pass to the underlying variations of mixed Hodge structure via André's fullness theorem.

Recently, a class of spin chains known as ``free fermions in disguise'' (FFD) has been discovered, which possess hidden free-fermion spectra even though they are not solvable via the standard Jordan-Wigner transformation. In this work, we extend this FFD framework to open quantum systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We establish a general class of exactly solvable open quantum systems within the FFD framework: if the Liouvillian frustration graph is claw-free and has a simplicial clique, the Liouvillian possesses a hidden free-fermion spectrum. In particular, the (even-hole, claw)-free condition automatically guarantees this, enabling exact computation of the Liouvillian gap and an infinite-temperature autocorrelation function. Our results provide the first realization of the FFD mechanism in open quantum systems.

We show that the Baum--Connes assembly map factors canonically through the unitary conjugation groupoid, which serves as a universal mediator among groupoid models that are Morita equivalent to a given transformation groupoid. This establishes a structural link between groupoid-based index theory and the Baum--Connes program at the level of K-theory. Building on our previous development of unitary conjugation groupoids and their associated index theory, we extend the $K_1$ index framework beyond the Type I setting to non-Type I examples, including the irrational rotation algebra and amenable crossed products. Using Morita equivalence, we relate unitary conjugation groupoids to transformation and action groupoids, enabling the transfer of descent-type index constructions to these settings. Our main result shows that, among all groupoid realizations that are Morita equivalent to a transformation groupoid, the factorization through the unitary conjugation groupoid is canonical at the level of K-theory. This identifies the unitary conjugation groupoid as a universal intermediary for the Baum--Connes assembly map. As applications, we recover the classical index pairing with the tracial state for the irrational rotation algebra in the sense of Connes, and we prove that for amenable crossed products the descent construction agrees with the analytic Baum--Connes assembly map under Morita equivalence. These results provide a conceptual interpretation of the assembly map in terms of internal symmetries of crossed product algebras and suggest a unified framework connecting Fredholm-type index data with equivariant K-theory via groupoid methods.

We give an expression for the determinant of the twisted Laplacian associated with any linear representation of a finite quiver in terms of traces of the holonomy of its cycles. To establish this expression, we prove a general identity for the determinant of a block matrix in terms of traces of products of its blocks. We give two proofs, one purely enumerative and one using generating series. In the special case of a finite graph equipped with a vector bundle and a connection, the twisted Laplacian determinant admits a combinatorial interpretation as a weighted count of tuples of oriented cycle-rooted spanning forests, where the weights involve traces of holonomies along cycles formed by combining the edges of the forests.

This paper addresses the topic of equidistribution and recurrence for polynomial sequences over function fields. The main focus is to note and correct two small errors in [V. Bergelson and A. Leibman, A Weyl-type equidistribution theorem in finite characteristic, Adv. Math. 289 (2016) 928-950], contextualized within the broader developing literature on number theory and additive combinatorics in function fields. Connected with the resolution of these issues, we also prove new results characterizing intersective polynomials in finite characteristic in terms of various algebraic, combinatorial, and dynamical properties.

In this paper, we prove the commutativity between the Pansu pullback of a smooth contact map between Carnot groups and the differentials appearing in the spectral complexes. As a direct application, we also present a way of "lifting" a Pansu derivative (viewed as a Lie algebra homomorphism) from Carnot groups to their central extensions.

In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal C$ and its associated inner $2$-minor ideal $I_{\mathcal C}$, we prove that $I_{\mathcal C}$ is a complete intersection if and only if $\mathcal C$ is a chessboard.

We consider the algebra of square matrices of bounded non-commutative (NC) functions over NC operator unit balls (unit balls corresponding to finite-dimensional operator spaces) and characterize cyclic matrix free polynomials with respect to the canonical weak-* topology. More precisely, we show that a matrix free polynomial generates a weak-* dense left/right ideal if and only if it is stable, i.e., non-singular at every point in the NC operator unit ball. To this end, we establish a version of the Neuwirth--Ginsberg--Newman inequality for stable matrix free polynomials. We combine our techniques with the theory of realizations to establish cyclicity of stable NC rational functions that are uniformly continuous across the boundary, and we recover known results about cyclicity of NC rational functions in the matrix-valued free Hardy space over the NC unit row-ball. Lastly, we introduce the NC parallel sum function: a stable NC rational function that is contractive over the NC bidisk, which cannot be extended uniformly across the boundary, and determine its cyclicity using properties of accretive operators.

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime conductor $q$, a positive proportion of the central values $L(1/2,χ)$ do not vanish as $q\to\infty$.