Papers

We introduce the FlexGrad optimizer, an adaptive gradient method that addresses a key limitation of classical AdaGrad-type methods: their inability to exploit correlations between gradients. By scaling steps using both cumulative gradients and cumulative squared gradients, FlexGrad preserves AdaGrad-like behavior in noisy regimes while allowing substantially larger steps when gradient directions are persistent. The method is fully online, horizon-free, and avoids any auxiliary search procedure during optimization. For convex Lipschitz objectives, we prove that FlexGrad attains the optimal asymptotic convergence rate for nonsmooth convex optimization, with guarantees for both global and coordinate-wise variants. FlexGrad is straightforward to implement and is closely connected to practical gradient-normalized methods. Experiments on a range of convex and nonconvex problems show that FlexGrad performs competitively across a range of optimization settings.

We study colored knot Floer homology via a bordered cabling construction. We construct a cabling module and identify the connecting maps in the defining direct system at the level of this module. By taking the colimit at the chain level, we obtain a general formula for colored knot Floer homology of an arbitrary knot in terms of its ordinary knot Floer homology and the colored unknot algebra. This extends the computation of Alishahi-Gorsky-Liu from L-space knots to all knots.

We construct a two-parameter family of polynomial vector fields in $\mathbb{R}^3$ and prove that, on a nonempty open set of parameter values, every member of the family is Hurwitz and possesses two non-constant periodic orbits, one of saddle type and one attracting. This answers Problem 19 in Gasull's list of open problems on low-dimensional dynamical systems concerning the existence of non-constant periodic orbits in Hurwitz smooth vector fields on $\mathbb{R}^3$. This also provides counterexamples to the expected non-existence of non-constant periodic orbits for Hurwitz polynomial vector fields; a question that arose from the first counterexamples to the Markus-Yamabe conjecture in the late 1990s and that was stated by Meisters in his featured MathSciNet review of the Bernat-Llibre counterexample.

We study the conjectural characteristic class analogue of the Hodge index theorem for singular complex algebraic varieties, formulated by Brasselet-Schürmann-Yokura which expresses the Goresky-MacPherson homology $L$-classes in terms of suitable Hodge-theoretic $L$-classes, for quotient spaces $X/G$, where $G$ is a finite group and $X$ is a pure-dimensional complex projective variety. Assuming that an equivariant $K$-theoretical version of the conjecture holds for $X$, we show that the characteristic class conjecture holds for $X/G$. Without assuming the equivariant $K$-theoretical version, we show that the characteristic class conjecture holds for $X/G$, provided that it holds for all fixed point sets $X^g$ and that a suitable normally nonsingular inclusion assumption is satisfied. Our approach is to work in equivariant analytic $K$-homology and equivariant algebraic $G$-theory and to identify the corresponding localized classes. On the analytic side, we identify the Banagl-Zagier equivariant $L$-classes with localized Chern character of the equivariant $K$-homology class of the signature operator defined by Banagl-Leichtnam-Piazza. On the algebraic side, we compute the localization of the equivariant motivic Hodge-Chern class transformation of the intersection Hodge module.

A conference signal is a complex-valued function on a finite abelian group that vanishes at 0 in both the time and frequency domains, and is otherwise flat in both domains. For example, the Legendre symbol is a conference signal on $\mathbb{Z}_p$ since it vanishes at $0$, takes values $\pm 1$ away from zero, and is a scalar multiple of its Fourier transform. More generally, any nontrivial multiplicative character on a finite field is a conference signal on its additive group. Until now, conference signals have mainly been studied by number theorists, who wanted to know if multiplicative characters were the only conference signals on finite fields. The state of the art uses fancy tools from symplectic geometry to prove that non-character conference signals exist on non-small fields. The same tools show conference signals exist on every abelian group of odd order, without saying what they are. In this (human-friendly!) paper, we explain how conference signals produce other objects of interest in algebraic combinatorics and applied harmonic analysis, and then we present some new, explicit constructions of conference signals on $\mathbb{Z}_n$. In particular, we give the first explicit conference signals for composite $n$ and the first explicit non-character examples for prime $n$.

Central limit theorems in survey sampling are typically established under specific sampling designs rather than from generic characteristics of a design. Motivated by the search for more general conditions, we investigate whether low-order inclusion probabilities can provide such a basis. First- and second-order inclusion probabilities determine the first two moments of sampling estimators such as the Horvitz–Thompson estimator, but we show that even exact agreement of these probabilities with those of simple random sampling without replacement (SRSWOR) does not preserve asymptotic normality. We construct a sequence of fixed-size sampling designs having exactly the same first- and second-order inclusion probabilities as SRSWOR, together with a sequence of finite populations for which the sample total converges to $N(0,1)$ under SRSWOR but to a centered standard exponential distribution under the alternative designs. Thus, the two sample totals have exactly the same mean and variance at every stage of the sequence, yet have different limiting distributions. The alternative designs are obtained from balanced incomplete block designs arising from finite projective planes. Their cyclic structure allows us to prescribe the distribution of the sample total under block sampling, while a probabilistic permutation argument, combined with an exponential inequality for sampling without replacement, yields finite populations satisfying the SRSWOR central limit theorem. The result shows that low-order inclusion probabilities do not control the global dependence structure relevant to asymptotic normality and therefore cannot, by themselves, provide general conditions for central limit theorems under complex sampling designs.

It is shown that the mapping class group of any non-sporadic surface admits a continuum of equivariant coarse median structures, thus improving a previous theorem of the author. The same holds for a wide class of right-angled Artin and Coxeter groups, including examples of any dimension at least two, and RACGs which are not quasi-isometric to RAAGs; the coarse median structures in these cases are not induced by cocompact cubulations. The results are instances of a general procedure, which uses quasicocycles to flexibly modify the structure of a (suitable) hierarchically hyperbolic group.

The Double-Partition Coloring Problem (DPCP) is a recently introduced generalization of several coloring problems on graphs and hypergraphs, including the Partition Coloring Problem, the List Coloring Problem, and the Conflict-Free Coloring Problem (CFCP). In this work, we further investigate the structural and computational properties of the DPCP and develop exact and heuristic solution approaches. We derive structural results that allow the detection of infeasible instances and the reduction of their size. We propose one-step and two-step heuristics for obtaining high-quality solutions. These procedures are incorporated into an enhanced branch-and-price algorithm based on a set covering formulation of the DPCP. The algorithm further includes new strategies for solving the NP-hard pricing problem. Computational experiments show that the proposed branch-and-price algorithm and a compact integer programming formulation complement each other: while the compact formulation is generally more effective on sparse instances, branch-and-price performs particularly well on denser and larger instances. In particular, these are the first exact algorithms to be computationally evaluated on various CFCP instances.

The following article attempts to define a new digital signature scheme based on a variant of the MinRank problem under appropriate parameters. First, the key generation process will be defined, followed by the message signing and verification processes. The speed of signing and verification will be analyzed, and finally, the signature will be evaluated in terms of security and robustness against known attacks, which will lead to a set of requirements that we will consider when selecting certain parameters related to the size and number of variables. It should be noted that this signature, although simple in design, can serve as the foundation for an entire family of digital signatures based on the central idea that inspired its creation.

In the context of locally convex vector valued integrable tensors on open sets of a $n$-dimensional manifold $M$ developped in our previous work \cite{sil09261621}, in this paper we construct a framework in which to address both a variational problem associated with a Lagrangian $\mathscr{L}$, and a conservation law on-shell associated with actions $\phi$ and $\omega$. $\mathscr{L}$ is a $C^{p}$-map valued in a quasi-complete Hausdorff lcs $\mathcal{Z}$ and defined on an open set $O$ of a normed space, $\phi$ is a $C^{p}$-action parametrized by an open $A$ of a normed space $T$, acting over the space $\mathfrak{Z}^{O}$ of fields valued in a normed space $Y_{0}$ and defined on $W$ domain of a chart $\xi$ of $M$; $\omega$ is a $C^{p}$-action on $W$ parametrized by $A$. Via a result in \cite{sil09261621}, we construct $\mathfrak{A}:\mathfrak{Z}^{O}\times A\to\mathcal{M}(W,\mathcal{Z})$ where $\mathfrak{A}(u,\eta)$ is the action functional associated with $\mathscr{L}$, $u$ and $\eta$. Here $\mathcal{M}(W,\mathcal{Z})=\mathcal{L}_{s}(\mathcal{H}(W),\mathcal{Z})$ is the lcs of measures on $W$ and valued in $\mathcal{Z}$. By letting $\mathcal{A}(u):A\to\mathcal{M}(W,\mathcal{Z})$ be such that $\mathcal{A}(u)(\eta)=\mathfrak{A}(u,\eta)$, we state that $\mathcal{A}(u)$ is differentiable off-shell, find its differential map off and on-shell. We establish that if the differential map of $\mathcal{A}(u)$ on $0$ is $0$, then $\mathscr{N}[u]$ is conserved on-shell, and if $p=\infty$, then the divergence of $\mathscr{N}[u]$ equals $0$, where the divergence of a lcs valued vector field is defined and analyzed in \cite{sil09261621}. Here $\mathscr{N}[u]$ is a $\mathcal{L}_{b}(T,\mathcal{Z})$-valued $C^{(p-1)}$-vector field of $M$ on $U=\xi(W)$.

We study the homogeneous Kuramoto model on random 3-regular graphs. A connected graph $G$ is globally synchronizing if, for almost every initial condition, the gradient flow converges to a fully synchronized state, in which all phases coincide. It has been conjectured by Bandeira [Oberwolfach Rep. 21 (2024) 1163-1226] that a random 3-regular graph $G_{2n,3}$ is globally synchronizing with high probability. We disprove this conjecture in a strong way: $G_{2n,3}$ is not globally synchronizing with probability tending to 1.

In one complex variable the Schwarzian derivative characterizes the Möbius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors–Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in $\mathbb{C}^n$. Starting from Oda's definition we construct a bilinear operator whose norm, taken with respect to the Bergman metric, is invariant under the automorphisms of the ball, and we show that this operator measures the extrinsic curvature of the preimages of hyperplanes. We study the linearly invariant family defined by a bound on this norm, we prove that a sufficiently small bound implies univalence, and we describe the extension of Ahlfors–Weill type, which takes values in the space of hyperplanes. The polydisk and convex mappings are also treated, and open problems and exercises are included.

This article analyzes the idempotence of Giuga numbers and Carmichael numbers. The first sections will lead us to define an arithmetic function through which we will find a new characterization of Giuga numbers. The following sections will lead us to generalize Carmichael numbers, finding numbers that are Fermat pseudoprimes in certain specific cases, and to formulate various statements and open questions regarding their properties. Finally, we will find a new alternative condition, based on idempotence, that a number must satisfy to be a counterexample to the Agoh-Giuga conjecture and which, therefore, presumably, no number will ever satisfy. We will almost always work under the premise that n is square-free, since both numbers always have that property, but in certain situations we will extend some of the results to the general case.

We study cluster recovery in a two-state Gaussian hidden Markov model in the slowly mixing regime, where the transition probability $\delta$ may vanish with the sample size. Temporal persistence creates long homogeneous segments and fundamentally changes the recovery problem relative to the i.i.d. Gaussian mixture benchmark. Rather than being purely adverse, label dependence provides exploitable structure: the statistical difficulty becomes concentrated near the rare transition points, giving rise to recovery regimes governed explicitly by their frequency $\delta$. We first characterize, up to universal constants, the oracle Bayes risk for offline and online clustering. Both problems exhibit a nonstandard polynomial risk regime, but the sequential constraint induces an additional logarithmic localization cost, yielding a genuine online/offline gap. We also study fixed lag prediction with delayed label feedback and identify the effective memory horizon beyond which past labels no longer improve the optimal rate. When the signal direction is unknown and possibly high dimensional, the problem is governed by the effective signal strength \[ r_n^2 = \frac{\|\theta\|^4/\sigma^4} {\|\theta\|^2/\sigma^2+d/n}. \] We derive minimax lower bounds and construct fully adaptive polynomial time procedures attaining the sharp recovery rates. In particular, almost full recovery is possible precisely when $r_n^2\gg\delta$. If $n\delta\gtrsim\log n$, adaptation to both the signal $\theta$ and the transition probability $\delta$ incurs no first order loss, and the sharp exact recovery threshold (under the Hamming loss) is \[ r_n^2 = 2\log(n\delta)\,(1+o(1)). \] Thus the effective complexity of the problem is governed by the number of latent transitions, rather than by the sample size itself.

We study the tropical octahedron recurrence on the lattice tetrahedron $\{(x_1,x_2,x_3,x_4)\in\mathbb Z_{\geq0}^4: x_1+x_2+x_3+x_4=n\}$ with an intrinsic double-hive boundary law. Let $M_1^{(n)},M_2^{(n)},M_3^{(n)}$ be independent standard $n\times n$ GUE matrices and, for fixed $\ell_1,\ell_2,\ell_3>0$, set $ X_r^{(n)}=\ell_r\sqrt n\,M_r^{(n)},\qquad r=1,2,3. $ The boundary law on the two upper panels is the Gibbs density on the cone of double hives whose four exterior sides correspond, in the prescribed orientations, to the spectra of $X_1^{(n)},X_2^{(n)},X_3^{(n)}$, and $X_1^{(n)}+X_2^{(n)}+X_3^{(n)}$. We then perform an octahedron-recurrence sweep to obtain the value at every lattice point of the tetrahedron. After scaling lattice positions by $n^{-1}$ and recurrence values by $n^{-2}$, we prove that the resulting random field converges uniformly in probability to a deterministic Lipschitz function on $\{(x_1,x_2,x_3,x_4)\in\mathbb R_{\geq0}^4: x_1+x_2+x_3+x_4=1\}$. For every strictly interior target lattice point we construct a canonical planar bipartite graph supported on a union of three triangular panels and prove an exact perfect-matching formula for the recurrence value. Its variable weights are explicit interlacing gaps from the minor processes of $X_1^{(n)},X_2^{(n)},X_3^{(n)}$, together with explicit seam and matching-independent terms. The asymptotic analysis of this formula yields a surface-tension variational characterization of the deterministic limit. This matching rule supplies, in the present four-simplex setting, the kind of higher-dimensional combinatorial formula identified as a major cluster-algebra problem by Henriques and Speyer.

We prove that there exists an absolute constant $c>1$ such that for every $n \geq 3$ there exists an $n$-dimensional real normed space $X$ for which every linear projection $P$ onto a hyperplane satisfies $\|P\| \geq c$. More generally, for every integer $k \geq 1$ there exists a constant $c_k>1$ such that for every $n \geq k+2$ there exists an $n$-dimensional real normed space $X$ such that every projection $P$ onto a subspace of codimension at most $k$ satisfies $\|P\| \geq c_k$. The proof is based on an idea of Gleit and Lazar, who established an analogous result for the space of real $n \times n$ matrices equipped with either the trace-class norm or the operator norm

We establish local asymptotic normality (LAN) for joint inference on scale and roughness in stationary Gaussian random fields observed on increasingly dense regular grids in a fixed domain. Scale and roughness are represented by the coefficient and decay exponent of the leading spectral tail. Under spectral regularity conditions, we identify the joint normalization governing the LAN expansion and prove consistency and asymptotic normality of the maximum likelihood estimators. For \(N\) observations, the estimation errors are of order \(N^{-1/2}\) for roughness and \((\log N)/\sqrt N\) for scale. After componentwise normalization, the errors converge to a perfectly correlated Gaussian pair. Knowing either parameter improves the convergence rate for estimating the other by a logarithmic factor. The analysis uses generalized locally Toeplitz (GLT) theory, for which we develop a truncation principle accommodating nonintegrable limiting symbols and quantitative bounds on the contribution from low frequencies. We verify the assumptions for Matérn and generalized Wendland covariance families, establishing uniform spectral bounds for the latter. We also show that consistency and the limiting distributions persist under covariance misspecification when the true and fitted spectra share the same leading tail and their difference decays sufficiently rapidly. This includes misspecification between Matérn and generalized Wendland models under explicit parameter restrictions.

Identification is the task of recovering the parameters of an unknown ground-truth model from sampled data. When parameters other than the ground truth induce the same output distribution, data alone does not provide enough information to recover the ground truth, and the model is thus called unidentifiable. We study the identifiability problem for hidden Markov models (HMMs): given an HMM, is it identifiable? Existing work on HMM identification establishes conditions under which the ground-truth HMM can be identified. However, most of these conditions are sufficient but not necessary, meaning that, when a model does not satisfy them, its identifiability remains inconclusive. We instead take a computational perspective: is there a sound and complete algorithm that decides whether a given HMM is identifiable, and if so, what is the complexity of this decision problem? We consider the decision problems arising from the various notions of identifiability in the literature, including deterministic, generic, global, local, state-permutation- invariant, and finite-alphabet identifiability. We show that all of these problems are decidable in PSPACE, via reductions to the theory of the reals at various levels of its quantifier-alternation hierarchy. We further show that the deterministic variants are already coETR-hard (and hence coNP-hard) for simply parameterized families.

Let $S_g$ be the closed oriented surface of genus $g \geq 2$, and let $\Gamma$ be a finite collection of closed curves on $S_g$ that fills, in the sense that its complement is a union of disks. The total number of double points of such a $\Gamma$ in minimal position is at least $2g-1$; we call $\Gamma$ minimally intersecting when equality holds. We prove that the number $N_g$ of mapping class group orbits of minimally intersecting filling curve systems satisfies $N_g \sim 16^g (2g)!/(64\pi\sqrt{2}\, g^3)$ as $g \to \infty$. We also study the single-curve subproblem, in which $\Gamma$ consists of a single component. Every such curve is in particular a filling system, so the corresponding count $N_g^{(1)}$ satisfies $N_g^{(1)} \leq N_g$, which improves the upper bound previously known for single filling curves. Finally, we compute $N_g$ and $N_g^{(1)}$ exactly for small genus and conjecture an asymptotic growth rate.

This work analyzes a framework for learning the solution operator, or equivalently, the Green's function, of a general self-adjoint second-order elliptic operator $\cL$. Subtracting the singular part of the Green's function, either known analytically or supplied by an explicit Laplace parametrix, recasts the learning of $\cL^{-1}$ as learning an $\cL$-harmonic extension operator on the boundary. We formulate the process as operator learning between reproducing kernel Hilbert spaces (RKHSs) to show that the kernel decomposition strategy raises the algebraic input rate by two orders and, under uniform analytic regularity of the induced boundary data together with compatible output spaces, upgrades the output rate from algebraic to exponential. We extend the framework further to semilinear problems via an iterative decomposition in which the only learned object remains a linear operator, ensuring that the improved linear rates translate directly to the nonlinear iteration. Numerical experiments are provided to validate the theoretical development.

We study the regularity of eigenfunctions of the Dirichlet Laplacian on domains with corners. The specific question we consider is whether the eigenfunctions can be smooth, e.g. trigonometric, on the closure of their domain. We determine how this regularity depends on the local geometry of the corner. Local asymptotic analysis provides detailed information about eigenfunctions near corners and allows us to quantify the Hölder regularity of the normal derivative. We demonstrate the limitations of some of the methods used to study these problems in finding a complete set of eigenvalues on polygons.

Decision-focused learning (DFL) trains predictive models for downstream optimization, but existing methods largely assume centralized data. In cross-silo settings, federated learning offers a natural alternative, yet standard federated methods optimize prediction over decision quality and do not address heterogeneity in downstream objectives or feasible sets. This heterogeneity is especially challenging for DFL because small perturbations in polyhedral problems can cause discontinuous changes in optimal decisions, destabilizing client updates and aggregation. We propose FedRSPO+, a heterogeneity-aware framework for decision-focused federated learning, built on RSPO+, a regularized predict-then-optimize surrogate that smooths the decision map through projection. We show that RSPO+ upper bounds decision error and regret for the regularized decision and, under exact regularization and consistent LP solution selection, for the original LP decision. We further derive cross-client heterogeneity bounds that depend on both objective and feasible-set heterogeneity, vanish at homogeneity, and require no strong convexity. FedRSPO+ uses an annealed, modular training procedure compatible with standard federated personalization and aggregation methods. Experiments on synthetic knapsack, shortest-path, and real-world energy pricing tasks compare against prediction-only federated learning and DFL baselines under varying heterogeneity and communication budgets. Results suggest that smoothing is a useful ingredient for stable collaborative decision learning and provide a heterogeneity-aware foundation for federated DFL.

We give a relative consistency counterexample to the conjecture of Inamdar and Rinot that every regular uncountable cardinal carrying no amenable C-sequence is ineffable in the constructible universe L. The existence of a $\Pi^2_1$-indescribable cardinal is equiconsistent with the existence of a $\Pi^2_1$-indescribable cardinal carrying no amenable C-sequence and not ineffable in L. For every ground-model C-sequence on a weakly compact cardinal $\kappa$, generalized Hechler forcing adds a club D with $D\cap\beta\subseteq C_\beta$ at stationarily many inaccessible indices. We use this capture theorem in an iteration of length $\kappa^+$ after Johnstone's local lottery preparation.

In neutral-atom quantum computers, atoms are moved to target positions along paths of empty positions, and a target position may be reserved for one species of atom. Motivated by this task, we introduce Single-Move Labeled Token Routing: every source and every target vertex of a graph is assigned a set of labels, and tokens occupy the sources. A solution consists of a matching that assigns each source to a compatible target (one whose label set intersects its own), a route for each matched pair, and a movement order in which, when a token is moved, its route contains no other token. The problem is known to be polynomial-time solvable when every source is compatible with every target, and $\mathsf{NP}$-complete on grid graphs when each source is compatible with exactly one target. We prove that the latter case remains $\mathsf{NP}$-complete on grids and on planar graphs of maximum degree four even when some solution has pairwise edge-disjoint routes. On trees, the problem is known to be $\mathsf{NP}$-complete even for maximum degree three. We study trees through the solution edge multiplicity, the largest number of routes of a solution sharing an edge, and the candidate edge multiplicity, the largest number of compatible pairs whose paths share an edge. We prove that on trees of maximum degree three, the problem is $\mathsf{W}[1]$-hard parameterized by a bound on the solution edge multiplicity, even when a movement order is given, and that on trees of unbounded degree, it is $\mathsf{NP}$-complete even when the candidate edge multiplicity is at most eight. We show that on trees the problem is fixed-parameter tractable parameterized by the maximum degree together with the candidate edge multiplicity, and also by the candidate vertex multiplicity, the same count at vertices. Unless $\mathsf{P}=\mathsf{NP}$, neither the maximum degree nor the candidate edge multiplicity can be omitted.

In this paper, we present JAXIGA, an open-source library for isogeometric analysis based on JAX. The library separates the construction of the mesh from the computations which are traced for compilation and differentiation. This allows geometry evaluation, assembly and the supported solvers to be combined in a just-in-time compiled program, for which first-order reverse-mode derivatives can be computed. We consider three numerical methods obtained from a single pointwise energy density. The Galerkin and energy-minimization methods use the same discrete potential and have the same stationary points, while collocation enforces the corresponding strong form. The sensitivities of a converged solution are obtained by solving one adjoint problem, and their accuracy depends on the tolerances of both the primal and adjoint problems. We also describe hierarchical extraction for local refinement of the variational methods and the use of smooth spline spaces for fourth-order problems. The numerical results are compared with analytical solutions and independent FEniCSx and JAX-FEM computations. For the reported GPU benchmark with 36,448 unknowns, assembly is twenty-four times faster than on the host CPU. When the conjugate-gradient solver is also run on the GPU, the total solution time is reduced from 31.8 to 1.38 seconds, and the result agrees with the direct solution to $5\times10^{-11}$. These results show that the choice of solver and the reuse of compilation are important for obtaining an overall reduction in computational cost.

Many machine learning algorithms rely on the Fourier transform, which is also the inner core of many analog computing devices and, specifically, of optical computing. The square of the modulus of the Fourier spectrum naturally maps onto pairwise interactions, as in the renowned Ising model, making it applicable to combinatorial optimization. Here we show how the nonlinear Fourier transform, related to integrable nonlinear partial differential equations, significantly widens the computational capability and also enables a wide variety of computing gates. We study specifically the case of the nonlinear Schrödinger equation; we introduce an input composed by “tokens”, i.e., batches of information encoded in a multi-step potential, and we show how the nonlinear spectral computing arising in the wave propagation not only generalizes combinatorial optimization to higher order problems, but also realizes boolean gates. In the weakly nonlinear regime, nonlinearity generates higher-order interactions as perturbative corrections to the pairwise terms. In the solitonic regime, the discrete nonlinear spectrum performs digital computation through soliton generation. By using four input tokens, we demonstrate the programmability of all 256 Boolean functions of three variables, including 3-SAT instances. Our results show that integrable nonlinear propagation provides an unprecedented single physical platform for higher-order and logical computation, opening the way to a rich variety of new algorithms for analog computing, machine learning, and classical and quantum information processing.

We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d \geq 3$. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of free-energy solutions, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a set of critical values: there is a critical value of a parameter pair in the system of equations for which there is a global-in-time energy solution and there exist blowing-up free-energy solutions under a criticality condition is violated for the parameter pair.

We study the eigenvalues of the random Laplacian matrix $D - A$, where $A$ is a Wigner matrix with sub-Gaussian entries and the diagonal matrix $D$ contains the row sums of $A$. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics; in particular, after the appropriate centering and scaling, the largest eigenvalue converges to the Gumbel distribution as the dimension of the matrix tends to infinity. This confirms, for general sub-Gaussian entries, a phenomenon the authors previously established only in the Gaussian case [Electron. J. Probab. 30 (2025), Paper No. 104], resolving a conjecture raised there. As a corollary, for an Erdős–Rényi random graph, we show the asymptotic fluctuations of the algebraic connectivity (Fiedler value) can be described in terms of the Gumbel distribution.

Häggkvist and Thomason proved that every $n$-vertex oriented graph with minimum semidegree at least $(\frac{5}{12}+o(1))n$ contains every orientation of a Hamilton cycle. Using Szemerédi's regularity lemma, Kelly later improved this to the asymptotically sharp bound $(\frac{3}{8}+o(1))n$. Taylor subsequently generalized Kelly's theorem by proving that every sufficiently large robust outexpander with linear minimum semidegree contains every orientation of a Hamilton cycle. We revisit Häggkvist and Thomason's proof and show that it can be modified to give a regularity-free proof of Taylor's theorem (and thus Kelly's theorem). Building on this, we also give regularity-free proofs of related results on Hamilton-connectivity and linkage in robust outexpanders. As applications of these results, we are able to replace the use of the regularity lemma in known results on arbitrary orientations of Hamilton cycles in $n$-vertex digraphs with minimum semidegree at least $\frac{n}{2}$ and in $n$-vertex digraphs with minimum total degree at least $(1+o(1))n$. The first step is to show that, in an $n$-vertex robust outexpander with linear minimum semidegree, two uniformly chosen disjoint sets of logarithmic size satisfy Hall's condition with high probability. The second is to show that robust expansion is inherited by uniformly chosen linear sized sets. In both cases, we use the graph-container methods of Kleitman–Winston and Sapozhenko to reduce the possible obstructions to a small enough family to permit a union bound.

Let $P_h$ be a semiclassical pseudodifferential operator on a closed Riemannian manifold $(M,g)$, with principal symbol $p$. We consider perturbations $P_h+h^\alpha Q$, with $0<\alpha\le 1$, where $Q$ is a semiclassical pseudodifferential operator whose symbol oscillates at scale $h^\beta$, with $2\beta <\alpha$. Assuming that the classical dynamics of the unperturbed system on a regular energy surface $p^{-1}(E)$ is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order $|\log h|$. This formula relates the distribution of energy levels of the quantum system near the energy $E$ to the periodic orbits of its classical counterpart. The perturbation $h^\alpha Q$ affects only the phases appearing in the trace formula.