Papers

We construct an explicit two-parameter family of complete, non-compact, three-dimensional, smooth steady gradient generalized Ricci solitons with $\mathrm{SO}(2)\times\mathbb{R}$ symmetry, providing a cylindrical counterpart to the spherically symmetric solitons recently found by Podestà and Raffero. The family is parametrized by a flux constant $k>0$ and a conserved quantity $\mathcal{C}\ge 0$. For $\mathcal{C}=0$, the asymptotic geometry exhibits power-law decay; for $\mathcal{C}>0$, the metric converges exponentially fast to a flat cylinder of finite radius.

We show that the lower bound in the majorizing measures theorem holds for a large class of random vectors. Specifically, suppose $X \sim \mu$ is a centered random vector in $\mathbf{R}^n$ with \[ C_{\mathrm{KL}}(\mu) = \sup_{\substack{\theta \neq \eta \\ \theta, \eta \in \mathbf{R}^n}} \frac{\mathrm{KL}(\mu_\theta \| \mu_\eta)}{\|\theta - \eta\|_2^2} < \infty, \] where $\mu_\theta$ denotes the law of the translate $\theta + X$. Then, for every nonempty, bounded $T \subset \mathbf{R}^n$, \[ \sqrt{C_{\mathrm{KL}}(\mu)}\, \mathbf{E}_\mu \Big[\sup_{t \in T} \, \langle X, t \rangle \Big] \gtrsim \gamma_2(T), \] where the righthand side denotes Talagrand's generic chaining functional. This result recovers, as a special case, the lower bound in the majorizing measures theorem for centered Gaussian processes. Our argument critically relies on the rate-distortion integral, recently introduced by J. Liu.

In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\equiv3\pmod 4$ the number $2\det[a_{jk}]_{0\le j,k\le (p-1)/2}$ is congruent to a square modulo $p$, where $a_{jk}=(\frac{j+k}{p})+(\frac{j^2+k^2}{p})$ with $(\frac{\cdot}{p})$ the Legendre symbol. We also prove that ${\rm per}[j^{k-1}]_{1\leq j,k\leq n-1}\equiv0\pmod n$ for any integer $n>1$ with $n\not\equiv2\pmod 4$.

These lecture notes provide an introduction to first-order optimization methods with a particular emphasis on stochastic gradient methods. We begin with deterministic gradient based methods for unconstrained optimization and study their convergence under standard assumptions such as smoothness, convexity, strong convexity, and the Polyak-Lojasiewicz condition. We then turn to stochastic approximation and stochastic gradient descent, motivated by empirical and expected risk minimization in machine learning. The main focus is on convergence theory: we discuss almost sure convergence and convergence in expectation, derive classical convergence rates, and present selected advanced topics, including almost sure convergence rates and variance reduction methods.

Motivated by recent developments in stochastic modeling of clock jitter in Analog-to-Digital Converters (ADCs) as autoregressive processes of order one (AR(1)), we study the density and stability properties of AR(1)-jittered sampling sets for Paley-Wiener signals. While the stability of sampling sets under i.i.d. perturbations of the integer lattice is classical, no analogous analysis was previously available for the correlated jitter observed in practice. We show that, despite having the correct asymptotic density both on average and almost surely, such sets almost surely fail to be stable sampling sets. We complement this negative result with a finite-dimensional analysis, providing explicit non-asymptotic conditions on the number of samples $ 2N+1 $, the jitter standard deviation $ \sigma_\xi $ and the confidence level $ 1 - \eta $ under which the smallest singular value of the $ (2N+1) \times (2N+1) $ jittered sinc matrix exceeds a prescribed threshold $ \epsilon $ with probability at least $ 1-\eta $, so that the matrix is well-conditioned with high probability.

The study on shift spaces in ergodic theory has extended beyond the classical setting, but there is room to discuss an extension of the Kolmogorov-Sinai entropy in the ergodic theoretical point of view. On the other hand, the rate distortion dimension recently attracted attention in mean dimension theory because it behaves like the Kolmogorov-Sinai entropy on dynamical systems in the “large" spaces in which the usual entropy is in general infinite. According to these background, we investigate the connection between the Gibbs measure on the product spaces and the variational principle based on the rate distortion dimension. Concretely we calculate the rate distortion dimension of the Gibbs measure for a Hölder continuous potential depending on the first two coordinates and it satisfies the simplest case of themodynamic formalism based on the rate distortion dimension: the extension of the maximal measure of topological entropy. Notably, to the best of our knowledge, this is the first study to introduce Gibbs measures into mean dimension theory, and this connection allows the mean dimension with potential to be computed more naturally through the notion of the zero-temperature limit, in the spirit of thermodynamic formalism.

Primitive positive constructions of first order structures have been shown to be a very useful tool in universal algebra for the study of constraint satisfaction problems. However, they seemed to be very rarely studied in classical algebra such as group theory. This paper fills in this gaps by looking at structures and obstructions based on permutation groups and giving a full classification in this sub-area. This special case is also very important for the generalization to all first order structures as every permutation group describes an easy-to-check necessary condition for the existence of primitive positive constructions, also between structures that are not at all linked to permutation groups.

Motivated by the vision of making sixth-generation (6G) networks sustainable, we study the sparse antenna/array activation problems in uplink cell-free massive multiple-input multiple-output (CF mMIMO) networks. We first develop an antenna-level optimal bilinear equalizer (OBE) weighting framework, in which each access point-user equipment (AP-UE) pair is assigned a matrix-valued long-term weight to shape the contribution of individual antenna elements, thereby generalizing the conventional large-scale fading decoding (LSFD) strategy from scalar coefficients to antenna-element-aware weighting. Building on this structure, we formulate sparse antenna activation as structured sparsity-inducing mean square error (MSE) minimization problems, and design four activation schemes at two granularities: antenna-level and array-level, each with UE-specific and network-wide (all-UEs) variants. The resulting convex problems are solved efficiently via the proximal method with closed-form group-wise updates, while the network-wide schemes are modeled through hierarchical sparsity and handled by a tree-structured proximal operator. Numerical results under correlated Rician channels and a detailed power consumption model demonstrate that the OBE weighting scheme consistently improves spectral efficiency over the LSFD, with gains increasing with the number of antennas. Meanwhile, the studied sparse activation schemes can achieve substantial energy efficiency improvement and power reduction with controllable spectral efficiency loss.

We study the Abel–Jacobi map over the twistor-$\mathbb{P}^1$ in the context of Scholze's geometrisation of the real local Langlands correspondence. In a similar spirit to a result of Fargues over the Fargues–Fontaine curve, we prove that pullback along the Abel–Jacobi map induces an equivalence on Picard groupoids and use this to recover local class field theory for archimedean local fields.

Motivated by the recent work of Li and Wang on parametric generalizations of Van Hamme's $(C.2)$ supercongruence in the $q$-setting, we establish $q$-analogues of a supercongruence related to Van Hamme's $(B.2)$ supercongruence, recently obtained by the authors. In particular, we derive parametric extensions of these $q$-supercongruences by constructing suitable pairs of hypergeometric functions through the $q$-WZ method.

We give a proof of the cosmological splitting conjecture of Robert Bartnik from 1988, which expresses the rigidity of the cosmological Hawking–Penrose singularity theorem. It states that a timelike geodesically complete, globally hyperbolic spacetime which has compact Cauchy surfaces and satisfies the strong energy condition must split isometrically as a Lorentzian product. Our methods combine the construction of global viscosity solutions to the Lorentzian eikonal equation by Zhu–Wu–Cui with our recently developed elliptic approach to the proof of Lorentzian splitting theorems in joint work with Braun, Gigli and Sämann, where we make use of the $p$-d'Alembertian operator for $p < 1$.

Computing the Lipschitz constant of the solution map of a multi-parametric quadratic program is important for the analysis of optimization-based control. This problem is governed by three factors: the parameter dimension, the number of decision variables, and the number of constraints. While empirical evidence has long suggested exponential complexity, a rigorous complexity-theoretic proof has been lacking. In this paper, we fill this gap by proving that this problem is not only NP-hard but also APX-hard. Furthermore, we reveal that: (a) the problem becomes polynomial-time solvable when the number of constraints or decision variables is fixed; and (b) both NP-hardness and APX-hardness persist even in the scalar parameter case. These results confirm that the complexity stems from the number of constraints and variables, rather than the parameter dimension. Numerical experiments further validate these theoretical findings.

A longstanding open problem is whether every Heyting algebra is the lattice of truth values (i.e., of subterminal objects) of some elementary topos. A positive answer is known for complete Heyting algebras (i.e., locales) via sheaves, and for Boolean algebras via a construction due to Peter Freyd. We extend Freyd's construction to all étale-finite Heyting algebras, in the sense of Evgeny Kuznetsov. These are the Heyting algebras satisfying a generalisation of the law of excluded middle relative to some finite Heyting subalgebra. For every étale-finite Heyting algebra $H$, we use Esakia duality to construct an elementary topos whose lattice of truth values is isomorphic to $H$, thereby extending the class of Heyting algebras for which a positive answer to the Heyting-to-topos problem is known. The toposes we construct are categories of certain compact étale spaces. As a consequence, they are finitely propositional: every object has a finite cover by subterminal objects. We show that a Heyting algebra occurs as the lattice of truth values of some finitely propositional topos if and only if it is étale-finite. This exhibits an obstruction to extending the use of compact étale spaces beyond the étale-finite case.

Extending recent advances on constructive quantum field theory, we study cumulants of one of the simplest matrix models in the regime of bounded rank. We analyze both ordinary cumulants and scalar cumulants, which arise from the Weingarten calculus and are essential to the topological expansion in quantum field theory. Our results are valid for arbitrarily large positive coupling and provide new instances where techniques such as the variational approach are applied to cu- mulants.

We present new short exponential sum approximations of length $N$ for $f_1(x)=\frac{1}{a+x}$ with $a>0$ on $[0, \infty)$ and for $f_2(x)= {\mathrm e}^{-x^2/2\sigma}$ with $\sigma>0$ on ${\mathbb R}$ with geometric error decay ${\rho}^{-2N}$ for user-defined $N \ge 2$ and $\rho >1$. The approximations are built over consecutive intervals $[b_j, \, b_{j+1}) \subset [0, \infty)$, $j \in {\mathbb N}_{0}$, with interval lengths that depend on $\rho$ and grow exponentially for $f_1$ and are equidistant for $f_2$. All parameters determining the exponential sum approximations on $[b_j, \, b_{j+1})$ are easily computed from the initial parameters on $[b_0, \, b_{1})$, ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of $f_1$ and $f_2$. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of $\log(x)$ on $[1,\infty)$ and of the error function $\mathrm{erf}(x)$ with predictable geometric error decay. Numerical examples for $N=8$ and $N=10$ clearly illustrate the theoretical error estimates.

We present and analyze an embedded Trefftz discontinuous Galerkin method for reaction-diffusion problems on anisotropic meshes. The method is constructed by imposing a relaxed local Trefftz condition via an embedding into a tensor-product DG space, yielding a reduced global system while preserving the approximation properties of the underlying high-order discretization. We prove stability and quasi-optimality on anisotropic, possibly curved, quadrilateral elements, and derive anisotropic a priori error estimates. Numerical experiments for $h$- and $hp$-refinement, including curved-domain examples, validate the theoretical results.

In this paper, we study the non-symmetric system of semilinear damped $\sigma$-evolution equations with different damping types, where two power exponents of nonlinearities belong to the critical curve, by using moduli of continuity in nonlinear terms. Our goal is to determine the sharp conditions on these moduli of continuity that guarantee the global (in time) existence of Sobolev solutions or, conversely, lead to finite-time blow-up. Furthermore, by employing the analysis introduced in the proof of our blow-up result, we provide a positive answer to an open problem for the symmetric models posed in the literature.

Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.

Formal rigor distinguishes mathematics from other disciplines, in the sense that mathematical statements are derived from explicit axioms by logically verifiable steps. Interactive theorem provers support this by expressing definitions, theorems, and proofs in a fully formal language and verifying them mechanically. We consider the benchmark problem of formalizing all published mathematics as a machine verifiable and continuously updated corpus of mathematical knowledge. This viewpoint treats mathematics as a structured database of interdependent results and raises questions about scalability and organization of large formal libraries. As a case study, we present an ongoing formalization in categorical algebra, namely dilatations of categories, extending classical localizations and illustrating what such an implementation looks like in practice.

It is well-known that the midpoint rule preserves the dissipation inequality if applied to a certain class of energy-based models. We introduce an appropriate scaling of the state variables such that the symmetric part of the resulting iteration matrix is guaranteed to be positive definite. This allows the application of three-term iteration schemes such as the methods of Widlund and Rapoport. Special emphasis is put on examples where the symmetric part is block diagonal such that the computations decouple. This then leads to efficient dissipation-preserving numerical schemes as illustrated in two numerical examples, namely the biharmonic heat equation and linear poroelasticity.

We uncover a finite-temperature extension of de Bruijn's identity – the classical relation $\frac{d}{dt}h(X+\sqrt{t}Z)=\frac{1}{2}J(X)$ connecting differential entropy and Fisher information. Our framework is the spectral theory of Blahut–Arimoto (BA) dynamics, recently developed by Wang~\cite{Wang2026} for the analysis of rate-distortion optimization. The central observation is elementary yet profound: for Gaussian sources, the spectral gap $\lam$ of the BA relaxation kernel $\G$ satisfies $\lam = 1/(2\beta\sigma^2)$~\cite{Wang2026}, while the Fisher information of the source is $J = 1/\sigma^2$. Hence \[ {\lam = \frac{J}{2\beta}} \] for all inverse temperatures $\beta > 1/(2\sigma^2)$. This identifies the BA spectral gap as a finite-temperature regularization of Fisher information. From this observation we derive an exact finite-temperature de Bruijn identity: \[ \frac{\partial F_\beta}{\partial \sigma^2} = \frac{1}{2\beta\sigma^2} = \lam, \] where $F_\beta$ is the BA free energy. This identity holds for all finite $\beta$ without any limit procedure. The classical de Bruijn identity follows as the exact consequence $\beta\,\partial F_\beta/\partial\sigma^2 = J/2$. The significance is structural: classical de Bruijn is not an isolated fact about Gaussian convolutions, but the $\beta\to\infty$ shadow of a one-parameter family of exact identities living in the spectral geometry of rate-distortion optimization. We discuss implications for the entropy power inequality, the $\chi^2$-dissipation structure of BA dynamics, and the geometric unification of information inequalities.

A finite group $G$ is $\varepsilon$-quasirandom if all its nontrivial irreducible complex representations have degree at least $|G|^\varepsilon$. Building on recent work of Golsefidy-Srinivas, we prove that expansion in a quasirandom group is controlled by expansion in its simple quotients. As a consequence, we remove the product theorem from the hypotheses of the Bourgain-Gamburd expansion machine. Moreover, we combine this result with crown theory to deduce that $1 + \lfloor \varepsilon^{-1} \rfloor$ random elements give an expander Cayley graph with high probability. Finally, generalizing results of Breuillard-Green-Tao and Pyber-Szabó, we prove that the diameter of any connected Cayley graph of a quasirandom group is polylogarithmic.

We study the tri-plane crossing number, that is, the minimal number of crossings in a tri-plane diagram for a bridge trisection of a knotted sphere in $S^4$. We show that every 2-knot in $S^4$ that admits a bridge trisection with at most five crossings is ribbon. As a consequence, we show that the 2-twist spin of the trefoil has crossing number 6. This is the first such computation for a non-trivial knotted surface.

Switched systems are a class of dynamical systems where trajectories switch between different systems based on a switching rule. This rule can depend on time and/or the position of the trajectory in the state space. The existence of a Lyapunov function implies the existence of a uniformly asymptotically stable equilibrium point at the origin, which we prove in this paper. A method to construct Lyapunov functions for switched systems using meshfree collocation and quadratic programming was described in a related article. We prove that, under suitable assumptions, the algorithm described in this previous work converges as the fill distance between the collocation points tends to zero.

A group is called realizable if it is the group of units in a ring with identity. The classification of realizable groups is a difficult open problem – originally posed by László Fuchs – and is an active area of research. Realizable groups seem rare, but their proportion within a fixed class of groups (cyclic, dihedral, finite abelian, etc.) varies. To quantify this proportion, we introduce the realizable density of a class of finite groups as an analog of natural density for subsets of the natural numbers. The realizable finite cyclic groups and the realizable finite abelian $p$-groups for $p$ odd have been classified; we prove that their realizable densities are 1/4 and 0, respectively. The realizable finite abelian 2-groups – and more generally the realizable finite abelian groups – have not been fully classified, and these special cases appear quite difficult. Nonetheless, we prove that the realizable density of finite abelian 2-groups is 1 and the realizable density of finite abelian groups is 1/2. Our work combines existing classification theorems for realizable groups with tools from analytic number theory.

A fermionic three-site generalization of the popular bosonic Swanson model is studied as providing an exactly solvable five-parametric example of the quantum-mechanical unitary-evolution process leading to an ultimate loss of the observability and fall in an exceptional-point singularity (EP3). The instant of degeneracy is found to have an explicit one-parametric form. Its unitarity-compatible vicinity (i.e., the corridor of access to EP3) is also specified in closed form. The exact, numerical-error-independent solvability is found essential due to another, avoided, false energy-level crossing which is found to occur not too far from the true EP3 singularity.

For fixed $\alpha>0$, we show that the sequence $\{\alpha n^{\theta}\}$ has Poissonian pair correlation for Lebesgue-almost all $\theta \in (0,\frac{3}{5})\cup(3,\infty)$. This improves a result of Technau and Yesha, who proved the same for almost all $\theta>7$. The approach of Technau and Yesha was based on a repulsion principle, which roughly allows one to estimate the variance of the pair correlation function using the fourth derivative of the phase. In our approach, we split the $\theta$-integration in the variance into many short intervals and show that most of the integrals can be estimated using the first derivative. The problem is then reduced to several counting estimates, which we prove using moments of the Riemann zeta function and exponent pairs.

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. The image of a non-constant morphism from a curve $T$ to $\mathcal{M}_g$ is a curve in $\mathcal{M}_g$. By work of González Díez and Harvey, for every integer $g \geq 3$, there exists a complete curve in $\mathcal{M}_g$. Here we generalize the construction to produce new complete curves in $\mathcal{M}_g$. We also find a formula for the genus of each curve $T$ using Galois theory for function fields.

We explicitly calculate Pan's geometric intertwining operator and the Fontaine operator on modular curves at infinite level via $q$-expansions, using Heuer's theory of cusps at infinite level. We prove that these two operators coincide on such expansions up to an explicit constant. As an application, we combine this result with $q$-expansion principles to provide a new proof of Pan's theorem that these operators are equal on the locally analytic vectors of completed cohomology of modular curves.

We prove results on moduli spaces of slope stable bundles of projective spaces on a hyperkähler manifold of Type $K3^{[2]}$. Let $X$ be projective of Type $K3^{[2]}$ and $h$ be a (generic) ample class. We prove that the moduli space $M_{{\overline{\bf w}}_a}(X,h)$ parametrizing $h$ slope stable bundles with a suitable mock Mukai vector ${\overline{\bf w}}_a$ contains an irreducible component $M_{{\overline{\bf w}}_a}(X,h)^{*}$ whose normalization $\widetilde{M}_{{\overline{\bf w}}_a}(X,h)^{*}$ is a (projective) HK manifold of Type $K3^{[a^2+1]}$, and that conversely every projective HK manifold $W$ of Type $K3^{[a^2+1]}$ is isomorphic to $\widetilde{M}_{{\overline{\bf w}}_a}(X,h)^{*}$ for a suitable $(X,h)$ as above. Moreover the universal bundle of projective spaces on $X\times \widetilde{M}_{{\overline{\bf w}}_a}(X,h)^{*}$ defines a vector bundle whose $2nd$ Chern class defines a rational Hodge isometry $H^2(X)\to H^2(\widetilde{M}_{{\overline{\bf w}}_a}(X,h)^{*})$. From this and a result of Markman one gets that the analogue of the Shafarevich conjecture (a special case of the Hodge conjecture) holds for rational Hodge isometries $H^2(W_1) \to H^2(W_2)$ between projective hyperkähler manifolds $W_1,W_2$ of Types $K3^{[a_1^2+1]}$ and $K3^{[a_2^2+1]}$ respectively. We prove results also for $(X,\omega)$ a general HK manifold of Type $K3^{[2]}$. In fact one ingredient in our proof is Verbistsky's theory of projectively hyperhomolorphic vector bundles.