Papers

We study a semimartingale optimal transport (SOT) problem where the cost depends on the full differential characteristics, and the minimisation is over all semimartingale laws with given marginals whose absolutely continuous characteristics lie in a prescribed closed convex set. Under only minimal assumptions of measurability, convexity, and lower semicontinuity on the cost function, we prove the existence of an optimal plan for SOT and establish a Kantorovich-type duality without time-regularity conditions. We further prove that SOT admits four equivalent formulations: (i) a Kantorovich duality formulation, (ii) a viscosity solution formulation of the Hamilton–Jacobi–Bellman equation, (iii) a martingale solution formulation via Markovian projection, (iv) a PDE formulation via weak solutions of a non-local Fokker–Planck–Kolmogorov equation. The framework simultaneously generalises classical optimal transport, martingale optimal transport, Schrödinger bridge problems, and barycentric weak optimal transport.

We study the discrete space-time wave equation on a $(q+1)$-regular graph $X$ associated with the affine Laplace-type operator. For the forward time-difference scheme we derive explicit formulas for the two fundamental solutions (wave kernels) in terms of discrete modified Bessel functions and the non-backtracking walk counts on $X$ thus providing a direct and explicit link between wave propagation and combinatorial graph data. Utilizing uniqueness property of the wave kernel, we prove a new trace-type formula associated to the affine Laplace-type operator on $X$ and apply it to deduce many combinatorial identities. For example, we derive a closed-form expression for evaluation of some trigonometric sums twisted by an additive character as well as evaluations of finite sums of Chebyshev polynomials twisted by binomial coefficients.

We develop criteria to distinguish the homotopy types of Gromov-Thurston manifolds. Our approach is based on a description of their fundamental groups as virtual Dehn fillings of relatively hyperbolic groups.

We study Kullback-Leibler (KL) contraction in finite Markov chains through a row-wise perspective. Evaluating the SDPI ratio at point masses yields a state-indexed retention profile $r(x)=D_{\mathrm{KL}}(P(x,\cdot)\|\pi)/\log(1/\pi(x))$ and a localization ratio $L(P)=\bar r_\pi/M\in[0,1]$ (with $M=\max_x r(x)$, $\bar r_\pi=\mathbb{E}_\pi r$) that distinguishes localized from global contraction obstructions. Our main contributions are (i) a convexity-gap identity showing that the gap between the row-averaged divergence and $D_{\mathrm{KL}}(\mu P\|\pi)$ equals the mutual information $I_\mu(X;Y)$, and a derived decomposition of the contraction ratio into entropy inflation and a mutual-information penalty; (ii) a Cheeger-type lower bound on $M$, tying the bottleneck geometry of $P$ directly to the row-retention profile; (iii) an explicit construction proving that $L(P_n)\to 0$ does not force $\eta_{\mathrm{KL}}(P_n)/M_n\to 1$, identifying cardinality of high-retention states (not their $\pi$-mass) as the decisive quantity. Alongside these, we record structural consequences, optimal Markov/reverse-Markov tail bounds for $r$, a Bhatia-Davis variance bound, two-sided spectral bounds with an explicit cubic correction, a KL/Pinsker mixing-time bound, and tensorization for product chains. We further show that $L(P)$ is structurally decoupled from the spectral gap, the Cheeger constant, and the mixing time: every vertex-transitive chain satisfies $L(P)=1$ regardless of its mixing speed, and the empirical rank correlations between $L(P)$ and these classical invariants on a diverse but limited test suite are essentially zero. The numerical experiments are exploratory and not used as evidence for a universal classification theorem.

The concept of graph capacity extends graph span by considering the maximum number of agents that can simultaneously traverse a graph while preserving a prescribed minimum distance. We study the Cartesian $1$-capacity, corresponding to the movement in which exactly one agent moves at each step. We establish general lower and upper bounds based on structural graph properties and derive exact results for trees. Our work highlights the role of branching and bridge structures in determining the Cartesian $1$-capacity and provides new insights into collision-free multi-agent motion on graphs.

The game of Cops and Robbers is a two player pursuit game on graphs where a team of cops attempts to catch a robber. The cop number $c(G)$ of a graph $G$ is the minimum number of cops needed to guarantee a winning strategy in $G$. A famous conjecture of Meyniel says that if $G$ is connected, then $c(G)=O(\sqrt{n})$. Erde, Kang, Lehner, Mohar and Schmid considered its generalization to $k$-uniform hypergraphs and conjectured that the cop number of such hypergraphs is $O(\sqrt{n/k})$. This may be understood as a hypergraph version of Meyniel's conjecture. In this paper we prove this conjecture for a class of expanding hypergraphs and show that with high probability the conjecture holds for random hypergraphs $H^k(n,p))$ provided $k\geq \log^3n$ and the typical degree, $p\binom{n-1}{k-1}$, is $\omega(\log^3n)$.

Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. For $\epsilon\in \{ \pm \}$, we define $\mathrm{cd}_{\epsilon}(G)=\{ \chi_{\epsilon}(1)\mid \chi\in \mathrm{Irr}(G) \}$, where $\chi_{+}(1)=\chi(1)$ denotes the degree of $\chi$, $\chi_{-}(1)=|G:\ker(\chi)|/\chi(1)$ denotes the codegree of $\chi$. Further, let $\omega_{\epsilon}(G)=\{ \pi(n)\mid n\in \mathrm{cd}_{\epsilon}(G) \}$, where $\pi(n)$ stands for the set of prime divisors of $n$. We established that if $|\omega_{\epsilon}(G)|\leq 3$, then $G$ is solvable. Additionally, a generalization of this result is obtained in the case when $\epsilon=+$.

For finite dimensional algebras over algebraically closed fields, we settle a question previously known only for certain families of algebras. More specifically, motivated by some foundational interactions between bricks and $\tau$-rigid modules, we show that a given algebra is brick infinite if and only if it admits infinitely many bricks which are not $\tau$-rigid. This proves the $\tau$-analogue of an open conjecture asserting that if (almost) all bricks over an algebra $A$ are rigid, then $A$ should be brick-finite. In retrospect, we strengthen some recent contributions to the study of a series of challenging open problems related to the $2$nd brick-Brauer-Thrall conjecture. Moreover, motivated by some of our arguments, we pose the question whether there exists any algebra that admits an infinite semibrick consisting of Ext-orthogonal rigid bricks. In connection with this, we present an algebra $A$ of rank $n$ that admits a semibrick of cardinality strictly greater than $n$ consisting of Ext-orthogonal rigid bricks.

We offer a workable construction of the free right nilpotent skew braces of arbitrary class which allows us to prove (among many other things) that this free object has free additive/multiplicative groups, and that it must also be residually finite and Hopfian. We introduce the class of right nilpotent solutions, which correspond to right nilpotent skew braces. As a consequence of our construction, the free solutions in this class have a solvable Word Problem, and every law holding for finite solutions of the previous type also holds for every solution of the same type. In the remainder of the paper, we present further explicit realizations of free objects and explore their consequences. Among these are free two-sided skew braces of abelian type (with an abelian multiplicative group) and free centrally nilpotent skew braces of class 2.

A.R. Camina and R.D. Camina posed in [CC06] the following question: Suppose there are two finite groups, one nilpotent and the other non-nilpotent, and the two groups share identical sets of conjugacy class sizes; must the non-nilpotent group possess a non-trivial center? Recently, W. Zhou [Zho25] gave a negative answer via a subtle and elegant construction of concrete counterexamples. Nevertheless, his approach relies on the existence of Sophie Germain primes, and thus fails to yield infinitely many counterexamples unconditionally. In the present paper, we construct an infinite family of counterexamples to Camina's question.

When implementing the method of multiple scales, one is traditionally instructed to treat the slow and fast time scales as if they were independent. Despite the intuitive motivation and the effectiveness of this perturbation method, one cannot failt to notice that these two scales relate to the same unique variable, so independence can only be formal. How sensible is it, then, to split a variable asymptotically into two (or more) independent ones? In this paper, we elucidate this issue with Van der Pol's equation, one of the simplest weakly nonlinear oscillators, as well as a simple example of a Hopf bifurcation. The discussion involves carrying the multiple-scale analysis up to arbitrarily large order and dealing with the divergent character of the resulting asymptotic series. Using the technique of optimal truncation, we re-connect the two scales. Specifically, we show that an initial translation of the fast coordinate leads to a non-trivial, exponentially small, phase shift that depends on the slow coordinate. This phase shift breaks the independence of the slow and fast scales and is found to result from the nonlinearity. Numerical simulations confirm its existence, as well as the predicted scaling. The calculation is carried out in sufficient detail to provide confidence in the generality of our result, both in its essence and in its form. In particular, we find strong indications that a Hopf bifurcation with a quadratic nonlinearity would lead to the same phenomenon, but with a larger magnitude.

Rémy's algorithm is a famous recursive construction of uniform random binary trees of growing size by a local grafting operation. In this work we construct a continuous version, a new local diffusion on the space of real trees of growing Brownian Continuum Random Trees (CRT's). It appears as the scaling limit of a variant of Rémy's algorithm due to Bacher, Bodini, and Jacquot. Once the trees are rescaled to have constant mass, this diffusion uncovers an ergodic dynamics on trees with the Brownian CRT as unique invariant law.

Let $C$ be the middle-third Cantor set and let $\mu$ be the natural Cantor probability measure. Let \[ \gamma=\frac{\log2}{\log3}. \] The two main results of this paper are \[ \mu\{x\in C:\|2^n x\|<n^{-\tau}\text{ for infinitely many }n\}=0 \qquad \text{ for } \tau>2-\gamma. \] and \[ \mu\{x\in C:\|2^n x\|<n^{-\tau}\text{ for infinitely many }n\}=1 \qquad \text{ for } \tau<\frac{1-\gamma}{2}. \] These results give new progress toward Velani's conjecture on zero-one law for dyadic approximation in the middle-third Cantor set.

We establish a local stability result for a class of Saint-Venant type inequalities. Given the solution $u$ of the Dirichlet torsion problem in a domain $\Omega$, we consider shape functionals $\mathcal{J}(\Omega)$ involving the integral of $j(u)$, where $j$ is convex and satisfies suitable structural assumptions. By Talenti's comparison principle, balls maximize $\mathcal{J}$ among sets of prescribed measure. We prove that this extremal property is stable in the class of nearly spherical sets: the deficit from the optimal value controls the square of the $H^{1/2}$-norm of the boundary perturbation. The argument relies on shape derivative techniques, including the computation of the second variation and the introduction of an adjoint state. As applications, the result covers several relevant examples, including the torsional rigidity, $L^p$-norms of the torsion function for $p\ge 2$, and Moser-Trudinger functional in dimension two.

In this paper, we introduce a new class of rings called Nil-$S$-Noetherian rings, which generalizes both $S$-Noetherian rings and $Nil_{*}$-Noetherian rings. We investigate several properties of this new class and establish generalized versions of some classical results, including Cohen's theorem, Levitzki's theorem, and Hilbert's basis theorem. Furthermore, we prove $S$-version of classical Lasker-Noether theorem for Nil-$S$-Noetherian rings.

We prove new logarithm laws for cusp excursions in spaces of lattices, and produce quantitative lower bounds for lattice points near submanifolds, using tools from dynamics and the geometry of numbers. As an application, we provide a new proof of power loss for the local Mizohata-Takeuchi conjecture with explicit error terms, as well as show that power loss is generic in $C^k$. The construction uses high-dimensional probabilistic estimates, but replaces the random orthogonal subspaces of Cairo-Zhang with random unimodular lattices; this yields stronger bounds and provides a richer family of counterexamples.

I provide a novel axiomatization of the Solovay model and a purely geometric treatment of the theory of balanced forcing.

We investigate spectral pairs associated with unbounded subsets of $\mathbb R^d$ that are periodic with respect to full-rank lattices. We establish a characterization of such spectral pairs in terms of the spectrality of an associated bounded fundamental component. More precisely, we prove that the Fourier transform defines an isometric isomorphism onto an $L^2$ space with an explicitly determined absolutely continuous pair measure, supported on a union of translates of a dual-lattice fundamental domain, if and only if the bounded component admits an orthogonal basis of exponential functions with frequencies in the dual lattice. In particular, the Plancherel identity alone is sufficient to characterize these spectral pairs. As an application, when the bounded component tiles $\br^d$ by a lattice, we obtain an equivalence between certain translational tiling properties of unbounded periodic sets and the existence of the corresponding Plancherel identities. We also examine periodicity properties of sets that tile by finite translation sets, highlighting distinctions between the one-dimensional and higher-dimensional settings.

We consider the sum of the adjoint Reidemeister torsions and prove the integrality for twist knots and the meridian. We also give some concrete examples of the generating functions for these sums.

The aim of this paper is to establish critical duality theory for ratios of nonnegative homogeneous convex functions (shorten for RC functions) and differences of convex functions (abbreviated as DC functions) on Banach spaces. Specifically, we establish a series of duality results on critical point theory and Morse theory for RC functions, including the homotopy type of sublevel sets, the Morse critical points and their Rothe critical groups, Lagrange critical points and their multiplicities, Lusternik-Schnirelman min-max critical values, Poincare polynomials, as well as the structure of handlebody decompositions, all of which are proved to be preserved under polarity dual. Moreover, we obtain the first critical duality theory of DC functions which does not depend on the DC decomposition. This answers a question left open from the work of Toland on DC functions and the work of Le-Pham on DC programming. We apply these results to provide a reformulation of the graph Cheeger constant using zonotopes; we introduce the contact data which serves as a geometric characterization of Lagrange criticality; and we show that the eigenproblems for 1-Laplacian and $\infty$-Laplacian on hypergraphs are equivalent to the contact problems of zonotopes, which indeed establishes a new characterization of zonotopes. We also prove a duality equivalence for certain nonlinear eigenvalue problems and bifurcation problems. Our study here reveals an intricate interaction of critical point theory with other fields such as convex analysis, combinatorial geometry, and nonlinear eigenproblems on graphs.

In this paper, we mainly study the interior $C^{2}$ estimates for a class of sum Hessian quotient equations on Riemannian manifolds. For $0\leq l<k< n$, we establish interior $C^{2}$ estimates at the center of a geodesic ball. Let $\Omega$ be a bounded domain (with smooth boundary) on Riemannian manifold. For $0\leq l<k\leq n$, we establish Pogorelov type estimates on $\Omega$ with the vanishing Dirichlet boundary condition.

Given an ideal in a number field, it is desirable in many situations to find two elements that generate the ideal over the ring of the integers of the field. Existing algorithms are either randomized, or impractical at cryptographic sizes. In the paper, we present a deterministic polynomial time algorithm to find the two-element representation of an ideal. For a monic irreducible integral polynomial \( f(x) \), let \( K=\Q[x]/(f) \) be the number field, and \( O_K \) be the integral closure. Our algorithm works when the norm of the input ideal is co-prime to the index \( [O_K:\Z[x]/f] \). In particular, it handles all ideals for monogenic \( f(x) \), a class that includes the cyclotomic polynomials widely used in lattice based cryptography. A key technical ingredient in our result is a generalized version of Dedekind criterion.

X-ray computed tomography reconstruction is an ill-posed inverse problem, particularly in low-dose and sparse-angle settings where measurements are noisy and incomplete. While learned reconstruction methods such as the Learned Primal-Dual algorithm achieve strong performance, they typically rely on supervised training with access to ground-truth data, which is often unavailable in practice. In this work, we propose a self-supervised reconstruction method by extending the Noise2Inverse framework to the Learned Primal-Dual algorithm. The resulting approach, called Noise2Inverse Learned Primal-Dual (N2I-LPD), enables training of a learned iterative reconstruction operator without ground-truth images by exploiting the statistical independence of noise in distinct measurements with respect to angular rotation of the CT-scan. We compare the proposed method with classical reconstruction methods, as well as neural network-based approaches such as a U-Net trained within the same N2I framework. The results demonstrate that N2I-LPD achieves improved reconstruction quality, highlighting the potential of combining learned reconstruction operators with self-supervised training strategies for practical CT imaging scenarios where ground-truth data is unavailable.

In this note we study smooth commutative group schemes over curves, whose generic fibre is an abelian variety. We prove a modified version of the conjecture proposed in [KS25].

Low-rank tensor approximation is a foundational tool for multidimensional data analysis in scientific computing, classically dominated by Tucker and Canonical Polyadic (CP) decompositions. While widely adopted, these standard approximation schemes represent data as sums of rank-1 tensors formed via mode-wise outer products. This inherent mathematical structure captures the independent variations of individual modes but systematically neglects the mutual interactions and coupled dimensional interdependencies natively embedded within the tensor. To overcome this structural limitation, we introduce the Holistic Multivariance Decomposition (HMD) framework. HMD provides a novel tensor decomposition algorithm that explicitly models both isolated mode effects and higher order mutual relationships through specialized projection operators. Numerical evaluations focusing on three distinct benchmarks from various fields demonstrate that the proposed HMD framework consistently yields significantly lower reconstruction errors compared to both Tucker and CP decomposition. These results establish HMD as a robust, high fidelity computational method for resolving complex, deeply coupled multidimensional data structures in science and engineering applications.

We find a normal form for the simplest Lagrangian singularity appearing as the projection onto the space-energy of a solution to the Hamilton-Jacobi equation. Using this normal form we approximate asymptotic solutions of an equation $(\widehat{H}-E_0) \, u_h = f_h$ with a semiclassical distribution $f_h$ microlocalized on a Lagrangian manifold $\Lambda_0$ when $E_0$ is a critical value of the restriction $H|_{\Lambda_0}$.

Let $h$ be a whole plane Gaussian free field, and let $\Omega$ be a bounded domain in two dimensions. We study the asymptotics as $t\to 0$ of the Liouville quantum gravity (LQG) heat trace, defined as the integral over $\Omega$ of the on-diagonal LQG heat kernel. Our main result is to show that the second term in the spectral expansion as $t\to 0$ of the expected heat trace is governed by a nontrivial exponent, given by the KPZ (Knizhnik–Polyakov–Zamolodchikov) relation. A similar but stronger (almost sure) result applies to the related notion of heat content. Along the way we obtain various results on the short-term behaviour of the heat kernel, notably solving a conjecture of \cite{BW} concerning its annealed asymptotics, and showing the finiteness of all moments of the properly rescaled heat kernel.

We study both the deterministic and randomised variants of noise-guessing decoding in additive memoryless channels. The error and complexity exponents of such decoding schemes are analysed under mismatched decoding metrics, and then specialised to matched, $\alpha$-tilted, and universal decoding metrics. The $\alpha$-tilted metric is proportional to the $\alpha$-th power ($\alpha>0$) of the true noise distribution. In deterministic decoding, the tilting operation does not affect the performance: all these metrics are equivalent to the matched one ($\alpha=1$), and are optimal for both average error and complexity. On the other hand, in randomised decoding, the matched metric is not optimal for complexity exponents; we show that the decoder needs to tune the parameter $\alpha$ according to the code rate in order to simultaneously achieve both optimal exponents using a decoding metric in that family. Finally, a universal decoding metric based on the empirical entropy of the noise sequence achieves both optimal exponents, independently of the channel law and uniformly over code rates, for the deterministic and randomised variants.

In this paper we develop fast numerical algorithms for solving shifted linear systems with semidefinite quasiseparable matrices. A combination of Givens and hyperbolic plane rotations is used to update the Cholesky-type factorization of the input quasiseparable matrix by determining a factorization of its shifted version of the form $LDL^T$, where $L$ is lower triangular and $D$ is a signature matrix. If the shifted matrix is also definite then the Cholesky factorization of the shifted matrix is computed in a stable way by using orthogonal transformations. Since quasiseparability is maintained under diagonal shifting, a fast variant of the updating procedure using computations with generators is also devised. Numerical experiments show the effectiveness and robustness of the proposed algorithm.

We prove the nonnegativity and palindromicity of a family of polynomials arising from $q$-Hoggatt numbers. The nonnegativity is derived from Stanley's $(P,\omega)$-partition theory through a standard Young tableau formula, while the palindromicity is proved by an involution on rectangular standard Young tableaux. Our result confirms a conjecture of Cigler.