Papers

For a numerical semigroup $S$ with embedding dimension four, we study the relationship between its type $t(S)$ and the cardinality of its minimal presentations $\eta(S)$. Using an approach based on the geometry of the Apéry set, we prove that $4t(S) + 5 \geq \eta(S) \geq t(S)-11$. This resolves a problem of Moscariello and Sammartano asking whether $t(S)$ is bounded by a function of $\eta(S)$, and improves the previously known bound $9t(S) + 4 \geq \eta(S)$ due to Bresinsky.

Let $1<p<2$ and $D_j$ be the discrete partial derivative on the Hamming cube $\Omega_n = \{ -1, 1\}^n$. Let $\Delta=\sum_{j=1}^nD_j$ be the discrete Laplacian. We prove the endpoint inequality \[ \left\|\left(\sum_{j=1}^n|D_jf|^2\right)^{1/2}\right\|_{p} \lesssim_p\|\Delta^{1/p}f\|_{p}, \quad \forall f:\Omega_n \to \mathbb C. \] This result answers the conjecture proposed by Naor, Eskenazis and Ivanisvili (see [BenEfraimLustPiquard, IvanisviliVolberg] or [Remark 45, EskenazisIvanisvili]). The proof relies heavily on the noncommutative semigroup BMO theory [JungeMei].

We study optimizing $L$-smooth and $\mu$-Polyak-Lojasiewicz (PL) objectives with unbiased stochastic gradients under $\alpha$-heavy-tailed noise ($1<\alpha\le 2$). In the unrestricted high-dimensional setting, we establish the noise-adaptive lower bound for attaining $\epsilon$-suboptimal function value. Under the generalized mirror-PL condition, we propose a centered-clipped mirror-descent method that achieves a high-probability upper bound matching the lower bound up to logarithmic factors. We further provide the improved tight stochastic complexity in fixed dimensions.

Motivated by the weak maximizing property ($\mathrm{WMP}$), we investigate $p$-summability conditions on maximizing sequences of bounded linear operators. Since the naive $p$-summability formulation is independent of $p$ and collapses to norm attainment for all operators, we introduce the weakly $p$-singular maximizing property ($\mathrm{SMP}_p$), based on maximizing sequences with no weakly $p$-summable subsequence. We completely characterize when the pairs $(\ell_p,\ell_q)$ and $(\ell_p,c_0)$ have the $\mathrm{SMP}_r$, revealing sharp contrasts with the $\mathrm{WMP}$. We also characterize the Schur property of order $p$ via the $\mathrm{WMP}$ and the $\mathrm{SMP}_r$. Under relative weak $p$-precompactness and suitable Dunford-Pettis-type assumptions, we characterize universal norm attainment for operators from $X$ to $Y$ in terms of either the $\mathrm{SMP}_q$ for adjoint operators or the weak$^{*}$-to-weak$^{*}$ maximizing property, and derive corresponding duality consequences for the $\mathrm{SMP}_p$. Finally, we introduce $p$-convergent perturbation properties for operators and their adjoints, characterize them for classical sequence spaces, and show that the $p$-convergent perturbation property is strictly weaker than the $\mathrm{SMP}_p$.

Here, we report high-order functional network connectivity as a promising way for studying the brain connectome. Traditional functional connectivity approaches capture only pairwise relationships between brain regions, overlooking complex multivariate dependencies that underlie cognition and behavior. First, we demonstrated that high-order interactions capture more information and can distinguish between resting-state and task-state brain activity. Second, we introduce a matrix-based entropy-functional method for estimating triadic interactions, which are statistical dependencies among triplets of brain regions, and apply it to large-scale functional brain networks. The resulting triadic networks revealed distinct community patterns that complement those observed in traditional pairwise functional connectivity analyses and simultaneously capture additional connection information. Despite the potential combinatorial explosion of triadic configurations, the networks exhibited constrained and hierarchical structures that allowed computation and interpretation. These findings position triadic connectivity as a promising next-step functional connectivity framework for probing brain network organization and high-order neural interactions, while also highlighting key biological and technical challenges that require careful consideration.

We study of the long-term behavior of a coupled wave-heat system. The system consists of a wave equation and a heat equation on two adjacent Lipschitz domains coupled by a common interface, with the heat equation being allowed to incorporate spatially dependent coefficients. We first establish strong asymptotic stability independent of the domains and coefficients. To this end, we employ the framework of closure relations, which reduces the spectral analysis of the coupled system to that of a wave equation. Secondly, we analyze non-uniform decay rates for classical solutions to the coupled system. Using a non-orthogonal decomposition of the state space, we reduce the problem to a resolvent-type estimate which is independent of the heat domain and heat coefficients. With this, we extend the known non-uniform decay rates to spatially dependent heat coefficients, yielding logarithmic decay under no assumptions and polynomial decay under the Geometric Control Condition.

Let $I\subseteq\mathbb C[x_1,\ldots,x_n]$ be an ideal generated by homogeneous polynomials of a common degree $d$. We give a polynomial partial differential equation criterion guaranteeing that $-n/d$ is a root of the Bernstein-Sato polynomial $b_I(s)$. We apply this criterion to the ideal of maximal minors of a generic $m\times n$ matrix and obtain the distinguished root $-n$; combined with local divisibility along determinantal strata, this allows us to obtain the strong monodromy conjecture in the maximal-minor case. Finally, we prove that the criterion is stable under enlarging the linear span of the generators, adjoining generators in disjoint variables, products satisfying the natural slope condition, and Thom-Sebastiani sums. These stability results provide new classes of homogeneous ideals and polynomials for which the distinguished Bernstein-Sato root can be detected. Keywords. Bernstein-Sato polynomial, monodromy conjecture.

We construct exact traveling-wave solutions for a semilinear parabolic equation with a specific nonlinearity. This nonlinearity arises, in particular, in voting models based on branching Brownian motion. Previously known results are special cases of those presented here.

We construct and characterize the one-parameter family of real-valued processes that allow to recover the law of a Brownian loop-soup of any intensity on the real line conditionally on its occupation time field. These processes generalize the Brownian burglar constructed by Warren and Yor (that corresponds to our process in the limiting case where the intensity vanishes and there is just one Brownian motion). These processes are closely related to the Bass-Burdzy flow and recent work of Aïdékon, Hu and Shi on the stochastic Jacobi flow. Our approach uses a formalism, building on the notion of driver processes, where one considers the evolving object to be the line that we “stretch", instead of the burglar itself and its remaining occupation time. This leads to a characterization of these (generalized) Brownian burglars by simple natural axioms. It also allows to treat “negative intensities” and provides fairly direct derivations of several properties of the burglars, and in particular a “target-independence/locality" property at the special intensity where the local time is the square of a Gaussian Free Field.

We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges $q_i$ of possibly mixed signs at distinct sites $\mathbf a_i\in\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\mathbf x):=\sum_iq_i|\mathbf x-\mathbf a_i|^{-3}$ does not vanish. Of course, for charges of one sign, $S$ is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of $S$.

In this paper we apply our work on polylogarithmic cocycles representing the Borel classes to establish that the comparison map from bounded continuous cohomology to continuous cohomology is surjective for connected semisimple Lie groups with finite centre. This proves conjectures of Dupont and Monod.

A classical theory of Amrouche, Girault and Giroire (1994) resolves the Laplace equation on $\mathbb{R}^n$ through an isomorphism between weighted Sobolev spaces. Inspired by their framework, we develop the corresponding $L^p$ theory for the fractional Laplacian: we introduce weighted fractional Sobolev spaces $\Lambda^{s,p}(\mathbb{R}^n)$, gauged at top order by $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, and prove that $(-\triangle)^{\frac{s}{2}}:\Lambda^{s,p}(\mathbb{R}^n)/\mathcal P_{[s-n/p]}\to L^p(\mathbb{R}^n)$ is an isomorphism for all $s\in(0,2)$ and $p\in(1,\infty)$; thus $(-\triangle)^{\frac{s}{2}}u=f$ is solvable for every $f\in L^p(\mathbb{R}^n)$, uniquely modulo an explicit finite-dimensional space of polynomials. More generally, extending the scale by duality to negative orders, the fractional Laplacians of the appropriate orders map its spaces isomorphically onto one another and compose exactly. A key ingredient is a family of weighted Hardy and Poincaré inequalities adapted to $\|(-\triangle)^{\frac{s}{2}}u\|_{L^p}$, established here for $1\le s<2$ by a reduction to the gradient. Since these inequalities hold a priori only on a dense class, while the density of $C_{\mathrm{c}}^\infty(\mathbb{R}^n)$ in $\Lambda^{s,p}(\mathbb{R}^n)$ is itself nontrivial, we first prove the isomorphism on the closure of the Schwartz class, which coincides with $\Lambda^{s,p}(\mathbb{R}^n)$ since the difference of two solutions is an entire $s$-harmonic function, hence a polynomial; the density follows as a by-product. As an application, we obtain existence and uniqueness, with explicit kernels and compatibility conditions, for the fractional Stokes system in $\mathbb{R}^n$.

Let $\mu$ be the Möbius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy–Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fejér majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.

The theory of combinatorial Hopf algebras (CHA) is a powerful framework for studying algebraic invariants of combinatorial objects. For instance, Stanley's chromatic symmetric function and the chromatic polynomial arise naturally from the CHA of graphs. In this paper, we define a CHA of signed graphs in order to study, in the same spirit, invariants of signed graphs such as the signed chromatic polynomial. The key ingredient of our construction is a well-chosen character, from which the signed chromatic invariants are obtained. Via the antipode, we obtain new proofs of combinatorial reciprocity results for these invariants. Moreover, by deforming the character with an additional parameter, we incorporate bivariate invariants into this framework, both for classical and signed graphs. In the classical case, this technique recovers the Tutte polynomial.

In this paper, let $u\in C^4(B_{10})$ with $D^2u\geq -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset\R^{n+1}$ satisfying \[ \sigma_2(\kappa[u])=f(x). \] We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality for \(b=\log(H+J_0)\), a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted $L^1$ estimate, which is completed using the Jacobi energy inequality and integration by parts.

Let $p$ be an odd prime and let $\mathbf{G}/\mathbb Z$ be a split connected reductive group with $\operatorname{dim} Z(\mathbf{G})\leq1$. Assume that a split maximal torus of $\mathbf{G}$ admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of $\mathrm{Lie}(\mathbf{G}^{\mathrm{der}})$ modulo $p$. We construct infinitely many continuous representations $\rho:G_{\{p\}}\longrightarrow\mathbf{G}(\mathbb{Z}_p)$ which are unramified at every finite prime different from $p$ and whose images contain a principal congruence subgroup of $\mathbf G^{\mathrm{der}}(\mathbb{Z}_p)$. When the center has dimension one, the representations may be chosen to have open image in $\mathbf{G}(\mathbb{Z}_p)$. The construction continues the author's earlier work on $\mathrm{GL}_n$-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For $\mathbf{G}=\mathrm{GL}_n$ all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd $p$ and every $n>1$, infinitely many representations $G_{\{p\}}\to\mathrm{GL}_n(\mathbb{Z}_p)$ with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction $p\geq7$ in that construction.

In this article, we introduce the concepts of generalized semi-Riemannian submersions and foliations, extending the classical framework to accommodate leaves with varying or degenerate causal characters, such as homogeneous foliations on semi-Riemannian manifolds and codimension-one lightlike foliations on Lorentz manifolds (including pp-waves). We establish that a regular foliation with a basic horizontal distribution is generalized semi-Riemannian if and only if it is transnormal. Furthermore, we introduce stationary foliations and prove that a stationary transnormal foliation induces a classical Riemannian foliation on the spatial rest space of a conformal observer. As a geometric consequence, we obtain a rigidity result ruling out stationary codimension-one lightlike foliations on positively curved Robertson-Walker spacetimes or static spacetimes. Finally, we derive an O'Neill-type sectional curvature formula for submersions with an involutive total distribution.

We propose a gradient mapping-type stationarity measure for the sum of two possibly nonsmooth nonconvex functions. Under suitable regularity assumptions, we show that Fréchet or proximal stationarity can be characterized through asymptotic vanishing of the proposed measure along some convergent sequence. We also establish analogous asymptotic results for a smoothing-based variant of the measure, which enables us to combine the proposed framework with smoothing techniques developed for nonsmooth optimization. As an application of our analysis of the stationarity measure, we provide an affirmative answer to an open question raised by [Olikier-Waldspurger, SIAM J. Optim., 2025] on whether every cluster point of a sequence generated by a proximal gradient method is a proximal stationary point under local Lipschitz smoothness of one component of the cost function. As a second application, we propose a proximal variable smoothing algorithm with a nonmonotone linesearch for minimizing the sum of two nonsmooth nonconvex functions under lower regularity of one component and prox-regularity of the other. For the proposed algorithm, we show that every cluster point of a subsequence such that the stationarity measure vanishes is a Fréchet stationary point.

In this paper, we show that $\operatorname{SL}_2(\mathbb{F}_{13})$ and $\operatorname{SL}_2(\mathbb{F}_{19})$ are Galois groups of totally real extensions of $\mathbb{Q}$. For each of these primes $\ell$, we find a polynomial of degree $\ell+1$ whose splitting field has Galois group $\operatorname{PSL}_2(\mathbb{F}_\ell)$. These fields satisfy Böge's criterion and the associated central embedding problem has a proper solution with Galois group $\operatorname{SL}_2(\mathbb{F}_\ell)$. One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang's modular equations for the Shimura curve $X_6^*(1)$.

We consider probability measures on compact subsets of the complex plane arising as limiting distributions of Galois conjugates of certain algebraic integers. These measures are characterized by infinitely many integral inequalities, one for each nonzero integer polynomial. We study a broad family of measures given by particular convex combinations of harmonic and equilibrium measures. We then show that, for measures in this family, it suffices to verify two conditions to guarantee all the required inequalities. Under additional arithmetic constraints on the underlying compact sets, we further show that the infinite collection of inequalities is equivalent to checking the inequalities associated with two explicitly specified integer polynomials.

We prove that the set of bounded ratios $\BR(X)$ on a semialgebraic set $X\subset\R^n_{>0}$ is the convex cone of linear forms that are nonnegative on the tropicalization $\trop(X)$. In particular, it is a rational polyhedral convex cone. For $X$ the set of Lorentzian polynomials with fixed M-convex support, it is the dual to the set of M-convex functions. We record an explicit counterexample to a conjecture of Huang–Huh–Soskin–Wang on the bounded ratios on Lorentzian polynomials. The bounded ratio in the counterexample corresponds to the non-hypermetric clique-web facet $\mathrm{CW}^1_7(1,1,1,1,1,-1,-1)$ of the cut cone on seven vertices.

For any link, we define a monodromy map from the motion group of the link to the group of automorphisms of the link's Khovanov homology. The image of this map is the unoriented monodromy group of the link. This map allows us to convert results concerning motion groups of links into ones about their Khovanov monodromy. In particular, we use a characterisation of the motion groups of split links to write their unoriented monodromy groups as an explicit semidirect product in terms of their unsplit pieces. We give some demonstrative examples, computing the unoriented monodromy groups of unlinks, Hopf links, and split links composed of pieces thereof.

In this paper, we study the combinatorial object $\bar{b}^k_{r,s}(n)$ which counts the overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. We extend results of Chacon and Sellers for several families of $r,s$ and $k$. We also establish divisibility properties for $\bar{b}^k_{r,s}(n)$ modulo prime $p$ and conclude with new congruences modulo powers of $2$. The techniques involved to obtain our results rely on theta function identities and modular forms.

We identify non-Gaussian corrections to the central limit theorem for the global and local laws, i.e. for Stieltjes transform of the empirical eigenvalue distribution of a large real symmetric or complex Hermitian Wigner matrix. We find that in the real case the rate of convergence in this CLT is substantially slower than in the complex case. For the proofs, we develop new static and dynamic hierarchies of extended cumulants up to any order that can be analyzed by a refinement of the zigzag strategy.

We prove an arbitrary-dimensional halfspace theorem for global minimizers of a smooth uniformly elliptic anisotropic perimeter: if the nonempty boundary of a minimizing set is contained in a halfspace, then the set itself is a halfspace. No evenness of the integrand or regularity of the minimizing boundary is assumed. The proof combines wall contact with a plane-peeling argument for the complementary phase defect and a simultaneous second blow-down. We also give a direct, self-contained exterior-barrier proof of the two-dimensional stationary anisotropic statement. Its rigidity conclusion is already covered by Bergner's earlier halfspace theorem; the proof is retained because it works directly with the anisotropic Euler-Lagrange operator.

The localized electromechanical phase transition in dielectric elastomers involves complex moving boundaries and severe electrostatic fringe fields driven by high-curvature interfaces. Traditional phenomenological models fundamentally underestimate the configurational forces by completely ignoring the in-plane electric field components and geometric singularities. Here, we present a asymptotically exact geometric framework to resolve the domain wall propagation. By mapping the highly deformed current configuration to a regular parametric strip via conformal mapping, the field singularities are algebraically eliminated. An exact integration by parts directly translates the higher-order geometric metric into a rigorous topological mass term, projecting the global conformal electrostatics into a non-linear $\sigma$-model Lagrangian density. Incorporating the Gent strain-stiffening model, the continuous translation symmetry yields a Hamiltonian first integral. Eigenvalue analysis and Bogomol'nyi-Prasad-Sommerfield (BPS) bound calculations prove that the domain wall emerges strictly as an asymmetric heteroclinic orbit connecting two saddle points. This geometric resolution provides exact analytical scalings for the localized interface energy, domain wall thickness, and asymptotic decay lengths, eliminating all phenomenological parameters and offering a deterministic paradigm for geometric instabilities in active soft matter.

A second maximal subgroup of a finite group $G$ is a subgroup $H$ that is maximal in every maximal overgroup of $H$ in $G$. We prove that every maximal subgroup of a prime-index subgroup of a finite group is a second maximal subgroup.

We develop a geometric framework for conservative mechanical systems modeled on conic pseudo-Finslerian manifolds, extending classical Riemannian and semi-Riemannian mechanics to anisotropic geometries. In this setting, we define a Jacobi-type pseudo-Finslerian metric and demonstrate that motions of fixed energy correspond, up to reparametrization, to its geodesics. We establish energy and momentum conservation under suitable symmetry assumptions via a generalized Noether's theorem, and provide new analytical criteria for the completeness of both the Jacobi metric and the mechanical system. Furthermore, for mechanical systems with holonomic and non-holonomic constraints, we prove the existence and uniqueness of reaction forces satisfying d'Alembert's principle. These results provide a unified geometric foundation for constrained and unconstrained anisotropic dynamics.

Two permutations $\sigma,\tau\in S_n$ are called disjoint if the composition $\sigma\tau^{-1}$ has no fixed point. If a family $\mathcal F\subseteq S_n$ contains no $s$ pairwise disjoint permutations, then a simple averaging argument gives $|\mathcal F|\leq(s-1)(n-1)!$. Inozemtsev, Kolupaev and Kupavskii characterized the equality cases in the range $s\leq n/(2^{17}\log n).$ We characterize all equality cases throughout the range $2\le s\le n$: equality holds if and only if $\mathcal F$ is a union of $(s-1)$ pairwise disjoint $1$-cosets. We also prove the linear statement underlying this classification: a real-valued function on $S_n$ has constant sum on every one-factorization if and only if it lies in the span of the indicators of the $1$-cosets. The proof is combinatorial and applies to every order, with a few small orders handled separately.

The way numbers are represented strongly influences which arithmetic structures are easy to see. The prime clockwork is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit mod or div operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=\nu_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this grand cycle is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.