Papers

Let $\Sigma$ be a surface of genus $g$ with zero or one boundary component and $n$ marked points, and $H$ a finite-dimensional factorizable ribbon Hopf algebra. We compute the central extension of the mapping class group of $\Sigma$, associated with a linearization of the projective representation constructed from the stated skein algebra of $\Sigma$ and $H$. Our proof is purely two-dimensional, and makes no use of TQFT arguments.

Topological defects play a central role in the mechanical behavior of crystalline materials, yet their relevance to amorphous solids has only recently begun to emerge. Over the last few years, theoretical, computational, and experimental studies have revealed the presence of well-defined topological invariants in vibrational eigenmodes, non-affine displacement fields, and deformation-induced vector fields of glasses. These defects have been shown to correlate strongly with soft spots, localized plastic rearrangements, yielding, and shear-band formation, suggesting a new perspective on the microscopic origins of plasticity in disordered materials. In this review, we provide a comprehensive overview of recent developments in the rapidly growing field of topological defects in glasses. We discuss the underlying theoretical concepts, including Burgers vectors, non-affine plasticity, vibrational modes, and topological invariants, and review recent numerical and experimental advances. Finally, we assess the current achievements, limitations, and open questions, and discuss future directions toward a unified topological description of plasticity and mechanical failure in amorphous solids.

We prove that every length space with curvature bounded from above that is geodesically complete has the Sobolev-to-Lipschitz property with exponent infinity. That is, every Sobolev map in the $W^{1,\infty}$-space has a Lipschitz representative so that the Lipschitz constant coincides with the infinity energy of the map. The proof is geometric and relies on arbitrarily small perturbations of geodesics to a curve that has zero length on the singular set. The motivation is to develop the analytic theory of such spaces; in particular, our result implies that GCBA spaces satisfy the infinity Poincaré inequality and an essential assumption in the theory of Lipschitz-Volume rigidity.

Khovanskii's theorem gives a Bezout-type upper bound for the number of isolated real solutions of a system of $n$ Pfaffian equations in $n$ variables in terms of three complexity parameters: the chain-degree $\alpha$, the degrees $\beta_i$ of the Pfaffian functions, and the order $s$ of the underlying Pfaffian chain. Despite its fundamental role in Pfaffian geometry and o-minimality, little is known about the sharpness of this bound. We investigate the theorem from a parameter-by-parameter perspective. We show that its dependence on the chain-degree $\alpha$ is asymptotically sharp by constructing, for every $\alpha,s \in \mathbb{N}$, a Pfaffian function of format $(\alpha,1,s)$ with at least $\alpha^s$ nondegenerate real zeros. We also show that its dependence on the degrees $\beta_i$ is asymptotically sharp: for fixed $n$ and $s$, we construct Pfaffian systems having $\Omega_{n,s}(\beta^{n+s})$ regular common zeros, matching the order of growth predicted by Khovanskii's theorem as $\beta\to\infty$.

A fundamental question in $p$-adic symplectic geometry and $p$-adic Hamiltonian mechanics is to what extent we can learn new information about integrable systems with coefficients in the rational numbers $\mathbb{Q}$, if we view them as $p$-adic systems instead of real ones. We combine geometric analytic estimates with Galois theory in order to show that at a local level, these real and $p$-adic viewpoints give completely different information. Then we focus on understanding this local behavior for the $p$-adic case by proving effective methods to compute local normal forms. Our results are motivated by a program outlined by Pelayo-Voevodsky-Warren a decade ago, to develop a $p$-adic theory of integrable systems and symplectic geometry.

Covariate selection in Generalized Linear Models (GLMs) is a fundamental problem in statistics, as including irrelevant predictors might lead to overfitting and poor interpretability, while omitting relevant ones might result in biased estimates. Most Bayesian approaches to variable selection – including spike-and-slab priors and continuous shrinkage priors – have key limitations, e.g., (i) are based on non fully conjugate formulations, (ii) are restricted to a linear model, or (iii) lack posterior consistency guarantees for the variable selection procedure and model parameters. In this work, we propose a fully Bayesian hierarchical and conjugate framework for covariate selection in GLMs, applicable to any distribution in the exponential family, based on modeling a binary inclusion indicator that directly encodes covariate inclusion in the linear predictor. In our approach, variable selection and parameter estimation are performed simultaneously, incorporating both sources of uncertainty in posterior inference. Consequently, our methodology provides a valid post-model Bayesian selection procedure. We present theoretical guarantees of the proposed fully conjugate Bayesian variable selection for GLMs, establishing posterior consistency of both the inclusion indicators and the active regression coefficients. We derive an efficient Gibbs Sampling algorithm with a corresponding R package implementation. We validate the proposed method on synthetic and real-world datasets, demonstrating competitive predictive and inferential performance.

We study a general class of self-repelling polymers on $\mathbb Z^2$, including the simple random walk, the self-avoiding walk and the repulsive Domb-Joyce model, in the presence of a constant force field acting on each monomer. Conditioning the polymer to have fixed length and fixed endpoints, we identify the limiting free energy and prove that typical trajectories concentrate exponentially near a deterministic macroscopic shape. This shape is characterized as the unique minimizer of a variational problem and can be interpreted as a geodesic of a height-dependent Finsler metric. We also analyze two limiting regimes with universal features: for small field strength, in the symmetric case, the geodesic is close to a classical catenary, while for large field strength it converges to a universal polygonal shape governed by the nearest-neighbor lattice constraint.

Motivated by the dimension-bound part of a problem of Bukh, we study Veronese circuits: how large can the span of $t$ linear forms be if their $m$-th powers are minimally linearly dependent? We prove the sharp finite dimension bound \[ \dim L\leq \frac{t+m-2}{m}. \] Here $\ell_1,\ldots,\ell_t$ are nonzero homogeneous linear forms over a field of characteristic zero, the powers $\ell_1^m,\ldots,\ell_t^m$ form a circuit, and $L=\Span\{\ell_1,\ldots,\ell_t\}$. Rational-normal-curve configurations attain equality for infinitely many pairs $(t,\dim L)$; in particular, the affine bound itself is sharp and the optimal leading constant in Bukh's question is $1/m$. The proof uses a coding-theoretic translation: the coefficient row space of the powers is the $m$-th Schur power of the coefficient code, and the minimality hypothesis makes this Schur power a full-support hyperplane to which the Schur-product Kneser theorem of Mirandola and Zémor applies. The same method yields flat-concentration and interpolation criteria, a Cayley–Bacharach lower bound, Segre–Veronese and positive-characteristic variants, and Hilbert-function constraints for equality and near-equality in Veronese circuits.

Classical kernel estimators of second order are interpreted in terms of random Weierstrass-type operators, particularly random Steklov operators. This leads us to obtain explicit nonasymptotic confidence bands and intervals for distribution functions $F$ and their derivatives $F^{(k)}$. Under the only assumption that $F^{(k)}$ is uniformly continuous, confidence bands for $F^{(k)}$ are established by using the Dvoretzky-Kiefer-Wolfowitz inequality. To give confidence intervals, we allow $F^{(k)}$ to have isolated discontinuities of the first kind, so that we really estimate the midpoint function $(F^{(k)})_{\star}(x)$. The proofs are based either on concentration inequalities for subordinated stochastic processes or accurate estimates of the MSE of the corresponding estimators. The length of the confidence bands and intervals depends on the degree of smoothness of $F^{(k)}$ measured in terms of the second modulus of continuity. Both lengths are of order $n^{-1 / 2}$ if $F$ is locally a polynomial of degree $k+1$ at most.

We establish Bareket's conjecture from 1977 for convex domains in all dimensions in the regime of weak boundary coupling. In other words, we consider the Laplace operator, subject to negative boundary conditions, and show that the ball maximises the first eigenvalue among all bounded convex domains of fixed volume, provided that the boundary parameter is sufficiently close to zero. The smallness depends on the volume and dimension only. The proof relies on a comparison with spherical shells with combined Neumann–Robin boundary conditions obtained via the method of parallel coordinates, which we manage to extend to all dimensions, and on a careful analysis of the corresponding radial problem.

We introduce the notion of tropical regularity as a generalization of tropical smoothness and present a geometric classification conjecture for tropical regular cycles. Roughly, the conjecture predicts that every regular tuple either forms a regular fibration or descends via a lattice projection to one of finitely many minimal-model cycles. Our main results establish this conjecture in three fundamental cases:unmixed regular tuples $(T,\ldots,T,[F])$ with $L(T)=\{0\}$ and $[F]$ spanning its ambient space; regular tropicalizations of complete intersections with generic coefficients; and regular tropical $2$-cycles satisfying the Hodge index inequality and spanning their ambient space.

Given a graph class $\mathcal{G}$, the limiting density of $\mathcal{G}$ is defined as $\delta(\mathcal{G})=\lim_{n\to\infty} \mathsf{ex}(\mathcal{G},n)/n$ where $\mathsf{ex}(\mathcal{G},n)$ is the maximum number of edges of a graph in $\mathcal{G}$ on $n$ vertices. The limiting density $\delta(\mathcal{G})$ is known to be a rational number when $\mathcal{G}$ is a minor-closed graph class. For every $\delta\in[0,\frac{3}{2})$, we prove that the set of $\subseteq$-minimal minor-closed graph classes with densities $>\delta$ is finite and we identify it completely. A consequence of our results is an algorithm that, given a finite set of graphs $\mathcal{Z}$, of total size $n$, either outputs the value of $\delta(\mathsf{excl}(\mathcal{Z}))$ or reports that $\delta(\mathsf{excl}(\mathcal{Z}))\geq \frac{3}{2}$, where $\mathsf{excl}(\mathcal{Z})$ is the class of graphs excluding the graphs in $\mathcal{Z}$ as minors. The algorithm runs in $2^{\mathsf{poly}(n)}$ time.

We develop a unified framework for the exact enumeration and asymptotic analysis of the three most studied classes of phylogenetic networks: tree-child (TC), reticulation-visible (RV) and orchard networks, whose cardinalities satisfy the strict ordering $|\mathrm{TC}_{\ell,k}|<|\mathrm{RV}_{\ell,k}|<|\mathrm{Orch}_{\ell,k}|$ for reticulation number $k\geq2$ (with $\mathrm{TC}\subsetneq\mathrm{RV}$ and $\mathrm{TC}\subsetneq\mathrm{Orch}$, while $\mathrm{RV}$ and $\mathrm{Orch}$ are incomparable as sets). Using the Chang–Fuchs structural theorem, we derive a two-level master functional equation for the RV bivariate generating function and obtain exact closed-form identities for the differences $\Delta_k(\ell):=|RV_{\ell,k}|-|TC_{\ell,k}|$ for $k=2,3$, with the asymptotic universality $\Delta_k(\ell)/|TC_{\ell,k}|\sim k!/\ell$. For orchard networks, we prove a universal hypergeometric law that resolves the exact enumeration problem for all $\ell$: the column generating function $F_\ell(v)$ is rational with denominator $D_\ell(v)=\prod_{j=2}^\ell X_j(v)$, where \[ X_\ell(v) = \sum_{k=0}^{\lfloor\ell/2\rfloor}(-1)^k\, \frac{\ell!}{(\ell-2k)!\,k!}\,v^k \] is the matching polynomial of the complete graph $K_\ell$ and a rescaled Jacobi polynomial. This immediately resolves the intractable $\ell=9$ case: $D_9$ has degree 20, dominant growth rate $\approx40.73$, and all spectral roots are positive real. A complete enumeration table is provided extending the published data of Cardona, Ribas and Pons.

Motivated by the desire to find a realistic and stable random model for $d$-dimensional signals, that are sparse in a transform-based and thus often coherent frame, such as a wavelet or a Gabor frame, we study the conditioning of incoherent sub-dictionaries sampled from a coherent dictionary, such as a unit norm frame. In particular, we show that if the sub-dictionary is selected via a coherence rejective Poisson sampling model, it is well-conditioned with high probability, as long as its expected size scales as $d/\log (K)$, where $K$ is the number of dictionary elements. The result is proved for the more general case of sampling quadratic sub-matrices from a real but not necessarily symmetric $K\times K$ matrix with zero diagonal, where coherence rejective sampling is defined via a symmetric mask, that acts as coherence substitute.

This paper presents a multi-objective perspective on block-structured integer programs featuring a single soft coupling constraint. By interpreting the coupling constraint as a second objective, we transform the coupled single-objective problem into an additively-separable bi-objective optimization problem. To avoid the expensive computation of the full Pareto front, we introduce an algorithm, which uses a binary search to isolate a region of interest around the soft constraint limit. This algorithm provides provable bounds on the single-objective optimum. We further enhance this algorithm, by exploiting the block-structure, using a novel $\lambda$-lookup mechanism to skip repeated sub-problem calculations. Finally, for scenarios requiring all non-dominated solutions within the region of interest, we propose a new approach, that works its way from the middle of the region of interest outwards. This algorithm shows quick convergence in terms of representation. Computational studies demonstrate that our methods dramatically reduce integer programming calls, thereby outperforming traditional dichotomic search. For large instances the method works as a strong heuristic providing bounds on the gap to an optimal solution, providing trade-off information in addition to the solution.

In this paper, we describe the 3D Nonconforming Virtual Element Method, where the local spaces are enhanced in order to build an $L^2$ projection of virtual functions onto polynomials of degree $k$. This higher order projection provides a more accurate representation of the load term and, more importantly, it enables the construction of the reaction term. We conduct a convergence analysis of the $L^2$ norm and $H^1$ seminorm errors with respect to mesh size and approximation degree. We also compare our results with those obtained using the conforming Virtual Element Method.

In this paper, we study quantitative homogenization for viscosity solutions of multi-scale semilinear second order partial differential equations (PDEs) on convex domains with Neumann boundary conditions. To this aim we use the probabilistic approach by studying the quantitative homogenization of backward stochastic differential equations (SDEs) associated with slow-fast systems of reflected SDEs.

Toric arrangements of maximal rank have been studied by the author in a paper that shows how the complement manifold of these arrangements is diffeomorphic to that of centered ones. In this work we turn our attention to toric arrangements of rank one, namely parallel toric arrangements. Our aim is to prove, by means of basic arguments of cohomology theory, that the Euler-Poincaré characteristic of the complement manifold of parallel toric arrangements can be computed in terms of those of the complement manifolds of the singular subtori that compose the arrangement.

Many operations decisions under distributional ambiguity, from pricing and inventory to capacity and contracting, evaluate an action through a tail-sensitive distorted utility of an uncertain payoff and hedge against the least favorable distribution consistent with a few known moments; the resulting worst-case evaluation is the inner problem of a moment-based distributionally robust decision. We study this inner problem, the extreme-case distorted utility under moment constraints, for a locally Lipschitz utility that may be nonsmooth and neither convex nor concave together with a general, possibly atomic, distortion. Recasting the problem in the quantile domain, we develop a unified method that yields exact first-order optimality conditions and closed-form extremal values and distributions for both the worst and best cases, drawing on nonsmooth variational analysis. A central step treats the monotonicity constraint by isotonic projection onto the monotone cone, turning an abstract infinite-dimensional restriction into an inexpensive inner solve that scales linearly in the discretization. The method recovers and extends classical moment bounds through three examples: a range value-at-risk extension of the Scarf bound, GlueVaR distortions with a reward–penalty utility, and a capped incentive contract under conditional value-at-risk. As the inner oracle of a robust min-max decision, the characterization embeds directly in outer robust optimization, illustrated on a real capacity-provisioning problem for generative artificial intelligence inference where accounting for moment ambiguity lowers required capacity while preserving service compliance.

We establish the higher differentiability of the local minimizers to a class of non autonomous convex integral functionals satisfying anisotropic subquadratic growth conditions, that include, as a particular case, those with orthotropic structure. The result is obtained under a gap bound on the exponents \(p_i\), that guarantees the local boundedness of the minimizers and under a suitable Sobolev assumption on the map that measures the oscillation of the energy density with respect to the $x$ variable, that is independent on the dimension.

We prove that any two Lévy processes on $\mathbb{R}^d$ with finite second moments admit a coupling that is itself a Lévy process and is both locally and globally optimal for the quadratic cost among all immersion couplings. This strengthens the optimality result of Kang and Lim (2025), which was established among Feller couplings. Our approach uses a martingale criterion for local optimality that applies more generally to Markov processes. The argument rests on the existence of an optimal Lévy coupling and a dual formulation of the associated optimization problem, both previously established by Kang and Lim (2025). We give simpler, self-contained proofs of these two results using optimal transport between infinitely divisible measures and Lassonde's minimax principle.

Consider a smooth one-parameter family of vector fields defined over some smooth manifold transitions from order into chaos. Inspired by the Second law of Thermodynamics, one is led to ask: can we find a flow whose dynamics realize this transition? To answer this question, motivated by the Mallet-Yorke Orbit Index theory, the Arnold-Khesin scheme for hydrodynamics and a heuristic argument by Rene Thom, we introduce a construction that transforms any one-parameter family of vector fields into a new object: the "Entropy flow". The Entropy flow is a flow defined on the product of the phase space with the parameter space and is best thought of as a flow generated by the original one-parameter family together with a drift in the parameter space, that pushes the trajectory of a given initial condition into a disordered, more complex state. To exemplify, for the Period Doubling, the Ruelle-Takens-Newhouse and the Intermittency routes to chaos the Entropy flow behaves exactly as expected - that is, it truly pushes trajectories into more complex states. In addition, in the spirit of Forcing Theory, in the paper we use the Conley index to discuss how one can use the Entropy flow to study the connection between topology and bifurcations. Moreover, drawing on the numerical and analytic evidence, we will analyze how the Entropy flow behaves in several examples of famous flows, including the Lorenz system, the Rössler attractor, and the breakup of the Shilnikov homoclinic scenario.

We prove that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if $(G,\cdot,\circ)$ is a simply connected Lie skew brace with nilpotent multiplicative Lie group and $\dim G>1$, then $(G,\cdot,\circ)$ is not simple. Thus, in the simply connected Lie setting, nilpotency of the multiplicative group is incompatible with simplicity in every dimension greater than one. The proof is carried out at the post-Lie algebra level. First, if the additive Lie algebra is solvable, then its nilradical is automatically an ideal of the associated post-Lie algebra. Second, when both Lie algebras underlying an integrable post-Lie structure are nilpotent, one always obtains a proper post-Lie ideal with trivial quotient. To pass from infinitesimal ideals to global ideals of the Lie skew brace, we show that trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals.

This paper focuses on a control-based approach for enhancing the regenerative capabilities of hydraulic multi-actuator systems using individual metering valves. Thanks to this architecture, pressure and displacement of each actuator can be controlled nearly independently. By determining online, the right pressure to be driven, it enables the optimization of regenerative control strategies for resistive or driving forces. Globally, this control strategy behaves such as a load sensing approach but each metering valve is piloted in order to activate regenerative mode when it is allowable. The main contribution relies on optimizing the pressure to be controlled in each actuator and the main pump in order to maximize the regenerative capacity of a hydraulic machine while following a displacement. The effectiveness of the proposed approach is proved in simulation. Only a single pump line regeneration is explored here but extensions to multi-pump or direct regeneration are also possible.

We consider the endomorphism tower of a monoid $M$, that is, the sequence of monoids End$_i(M)$ where End$_0(M)=M$ and for all $i\geq 1$, End$_i(M)$ is the monoid of all endomorphisms of End$_{i-1}(M)$. We show that for a finite monoid $M$ this sequence does not stabilise in a finite number of steps. Our focus is then on the case where $M=\mathcal{S}_n$, the symmetric group on a finite number $n$ of points. It is well known that other than in exceptional cases (which are avoided by taking $n \geq 7$), the corresponding automorphism tower of $\mathcal{S}_n$ stabilises at the first step. In spite of the natural nature of this question, nothing was known of the endomorphism tower above the level $i=1$. We determine (for each $n \geq 7)$ the elements of End$_2(\mathcal{S}_n)$ and their multiplication and thus verify that the monoids End$_i(\mathcal{S}_n)$ for $i=0,1,2$ all have group of units isomorphic to $\mathcal{S}_n$. We show that the same is true of End$_3(\mathcal{S}_n)$.

We classify monic finite-band scalar difference operators with independent coefficients admitting infinitely many support-preserving flows. We prove that such operators are completely characterized by an arithmetic condition on their support: the exponents must form an arithmetic progression. Conversely, every arithmetic support gives rise to an infinite hierarchy of local Lax flows. As a consequence, finite-band scalar Lax hierarchies with independent coefficients are classified by three integers (N,p,m), corresponding respectively to the leading order, the common difference of the support, and the number of generators. This framework recovers several classical systems, including the Toda, Volterra, Narita–Itoh–Bogoyavlensky, and Blaszak-Marciniak lattices, while simultaneously producing infinitely many additional examples. In particular, the support (-1,1,m) yields a scalar difference Lax representation of the Beffa-Wang hierarchy, and its Belov-Chaltikian reduction in the case m=2.

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.

Let $G$ be a group. The commuting graph $\zeta(G, V)$ of a group $G$ has a vertex set $V\subseteq G,$ two vertices are connected with an edge if the corresponding elements commute in $G.$ In this article, we study the commuting graphs for the non-abelian $p$-groups of order $p^n$ with center size $p^{n-2}.$ We study detour distance, metric dimension, resolving polynomials, and the spectral properties for these graphs.

Whether or not the 4-move is an unknotting operation remains an open problem. In this paper I show that every knot can be reduced to one with a trivial Alexander polynomial via a sequence of $4$-moves and isotopies.

Instantaneous Quantum Polynomial-time (IQP) circuits are promising candidates for near-term quantum computing due to the conjectured classical hardness of their sampling task and their shallow, highly parallelizable structure. Despite extensive studies of their complexity-theoretic properties, the role of circuit connectivity in their trainability as variational ansatze remains poorly understood. In this work, we present an investigation of IQP circuits for Hamiltonian optimization that traces a mechanistic chain linking circuit connectivity, entanglement generation, and gradient scaling: increasing connectivity drives the circuit output toward Haar-like statistics and volume-law entanglement, and this increase in entanglement is consistently associated with an exponential decay of the gradient variance and degraded optimization performance, while sparser architectures retain stable gradients at the cost of global expressibility and entanglement capacity. This connectivity-entanglement-gradient mechanism produces a nontrivial connectivity-trainability trade-off, demonstrating that effective IQP-based Hamiltonian optimization requires a balance between global expressibility and trainability rather than maximal expressivity alone. In particular, on our benchmarks, a circuit generating no entanglement already matches the optimization performance of more highly connected circuits, and a circular topology retains entanglement capacity while preserving trainability, making it a broadly effective choice among the architectures considered.