Papers

Over any local field of characteristic zero, we develop a local theory of Jacquet-Shalika integrals that were originally introduced by Jacquet and Shalika to study exterior square local $L$-factors for $GL_m$. In particular, at least in the archimedean case, we establish a complete functional equation matching Artin local $\varepsilon$-factors. Central to the argument is a novel comparison strategy between Jacquet-Shalika integrals and open-orbit integrals associated to the Shalika subgroup.

In this paper, we relate multiplicities of Bernstein–Sato-type polynomials with respect to a holomorphic function f to several singularity invariants. First, we introduce certain “mod” b-functions and show that they characterize the weight filtration on the localization of a simple regular holonomic D-module along f, and we provide an algorithm to compute them. Second, we show that they can be approximated by the multiplicities of roots of power b-functions (the b-functions with respect to powers of f). Further, we give a sharp upper bound for the Hodge level of elements given by a certain sum of such multiplicities. Next, we give an effective asymptotic solution to the Gelfand problem by determining an explicit threshold after which every integer shift of a root of b_f(s) is a pole of the Archimedean zeta function of f. We also show that the order of these poles is equal to the nilpotency index of the logarithmic monodromy operator, which we further express as the limit of the multiplicities of roots of power b-functions. We define several filtrations, relating them to the weight and Hodge filtrations, based upon which we leave some open questions that we address in the affirmative in the case when f has a homogeneous isolated singularity, or it is a hyperplane arrangement, or it is a semi-invariant on a spherical variety. We give several immediate applications to our results, including a positive answer to a question of Torelli assuming the hypersurface has log canonical singularities: 1/f lies in the intersection complex of the hypersurface of f if and only if -1 is a simple root of b_f(s).

We explain a general procedure to compute characters of chiral algebras of 2d (S)CFTs geometrically from their coadjoint orbits, in analogy with the coadjoint orbit method for finite-dimensional Lie groups. We demonstrate that this method can be used to give concise rederivations of the characters of the Virasoro algebra, Kac-Moody algebras, as well as supersymmetric generalizations.

We show that the derived categories of smooth families of twisted generalized flag varieties admit semiorthogonal decompositions into derived categories of twisted sheaves over the base. In particular, we obtain a categorification of Panin's computation of the Quillen K-theory of twisted flag varieties. The main ingredient is a generalization of the Samokhin-van der Kallen semiorthogonal decomposition for representation categories of parabolic subgroups from split simply connected semisimple groups to arbitrary quasi-split semisimple groups, including non-simply connected cases.

We consider the problem of establishing limitations on the power of quantum query algorithms via the completely bounded polynomial method. In particular, we prove several optimal functional inequalities involving different notions of completely bounded polynomials. These inequalities lead to limiting theorems for the power of quantum query algorithms that improve on prior works. 1. An optimal root-influence bound for block-multilinear polynomials. Prior work showed that block-multilinear polynomials $p$ of degree $t$ satisfy a root-influence bound, $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t^2$, which is stronger than the bound appearing in the Aaronson-Ambainis conjecture. We find the optimal constant in that inequality: $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t$. Since the amplitudes of quantum algorithms that query disjoint blocks of inputs-such as $t$-fold forrelation- are block-multilinear polynomials with $\|p\|_{\text{cb}}\leq 1,$ our inequality shows that they satisfy $t\geq \sum_i\sqrt{\mathrm{Inf}_i[p]}$. We prove that this inequality yields both a more efficient classical simulation than prior results based on the Aaronson-Ambainis argument, and a qualitative improvement: all classical queries are nonadaptive. 2. Optimal Fourier growth of the highest level of quantum query algorithms. We show that for every polynomial $p$ defined on $\{-1,1\}^n$ of degree $2t$, the Fourier Growth at the level $2t,$ namely $\|\widehat p_{2t}\|_{\ell_1}$, satisfies $\|\widehat p_{2t}\|_{\ell_1}\leq (en/(2t-1))^{\frac{2t-1}{2}}\|p\|_{\text{cb}}$. This is optimal up to the factor $e$, as witnessed by $2t$-fold forrelation. As quantum query algorithms that make $t$ queries (to the whole input) satisfy $\|p\|_{\text{cb}}\leq 1$, this yields a Fourier growth bound for these algorithms, partially resolving a question by Girish (STOC, 2026).

We study the roots and the Mahler measure of the $Q$-polynomial of links. We first consider links obtained by adding twists to a pair of parallel strands. We prove that the Mahler measure of the transformed $Q$-polynomial converges as the number of twists increases. We also show that all but a uniformly bounded number of distinct roots of the $Q$-polynomial approach the real interval $[-2,2]$. This behavior is different from that of the roots of the Jones polynomial under twisting. Numerical experiments on prime knots lead us to a conjecture about the roots of the transformed $Q$-polynomial of alternating knots. Finally, we compare real and unit-circle roots of the Alexander, Jones, and $Q$-polynomials, and give an infinite family of $2$-bridge links whose $Q$-polynomials have only real nonzero roots.

We associate normal toric transverse analytic envelopes with the singular orbits of diagonal holomorphic actions, and construct their canonical flat transport. At independent singular supports, the semiglobal first-integral algebra is determined by periodic resonances; complete integrability is equivalent to full resonance rank and finite envelope monodromy. We classify labelled residual actions with a fixed envelope by Grassmannian families and reconstruct the ambient weights from the labelled quotient and logarithmic residues at full resonance rank. For commuting vector fields in convergent simultaneous semisimple normal form, we identify the algebra of holomorphic first integrals with the common kernel of the derivations induced on the toric envelope. Under compatible flat transport this yields a semiglobal criterion. Polynomial examples show that losses of first integrals may be invisible to any prescribed finite jet.

In the Pearson random walk, the direction of the i-th step is a random variable uniformly distributed on the unit sphere of R^d, and its length is a non-negative random variable. We prove that, when d goes to infinity, the Euclidean norm of the walk converges almost surely to the Euclidean norm of the vector of step lengths, whose joint law is arbitrary. Further, we investigate the same convergence result in a more general framework, where both the number and the lengths of the steps depend on the dimension. The results in the present paper can be applied to approximate the distribution of some classical Pearson random walks, for example, in the Dirichlet case.

We show that the Navier-Stokes equations on $\mathbb R^2\times\mathbb T$ are globally well posed for initial vorticities that are both helically symmetric and integrable. This class of data typically generates velocity fields of infinite kinetic energy, so this result is not covered by the classical finite-energy well-posedness theory. This result is supercritical from the perspective of the three-dimensional Navier-Stokes scaling, and it is made possible by the special structure of helical flows using time-weighted Kato-type spaces.

Quandles can be regarded as generalizations of symmetric spaces. In the theory of symmetric spaces developed by Chen and Nagano, there is an interesting relationship between the two-number and the Euler characteristic. The two-number is a Riemannian geometric invariant that can also be characterized in terms of point symmetries, whereas the Euler characteristic is a topological invariant. The aim of this paper is to initiate the study of quandle analogues of Chen–Nagano theory. In particular, we investigate relationships between the two-numbers and the Euler characteristics of quandles, and provide examples of finite quandles that either satisfy or fail to satisfy properties analogous to those of symmetric spaces. These examples are constructed from directed simple graphs labeled by abelian groups.

We consider (feed-forward) Neural Networks (NNs) for the emulation of the solution to singularly perturbed second order boundary value problems, with two small parameters. We describe a shallow NN which exploits available asymptotic expansions for the solution. These additive decompositions into smooth and layer components, allow for derivative estimates which are explicit in the order of differentiation as well as the singular perturbation parameter(s) \cite{melenk, Irene, SX}. Utilizing such decompositions, we propose a simple NN for emulating the solution to such problems using the $\tanh$ activation function together with different training objectives, such as residual or energy minimization. The key idea is to augment the approximation space with suitable exponential functions, similar to enriched spaces in finite element methods, e.g.~\cite{Kellogg}. Numerical examples in one and two dimensions, including a smooth non-tensor-product domain, illustrate the resulting parameter-robust behavior over the tested perturbation ranges.

This paper establishes the exponential $L^2$-stability of the unique solutions to initial-boundary value problems for linear parabolic partial differential equations with general drift and zero-order coefficients in bounded domains. The key idea lies in constructing a suitable Dirichlet form with respect to a weighted measure $\mu = \rho\,dx$ and identifying the corresponding sub-Markovian $C_0$-semigroup of contractions on $L^2(U, \mu)$ with the unique weak solution. Remarkably, the exponential $L^2$-stability remains valid even when the zero-order term vanishes, and it holds robustly for all drift coefficients $\mathbf{H} \in L^p(U, \mathbb{R}^d)$ with $p \in (d, \infty)$.

Motivated by the strong cosmic censorship conjecture in general relativity, we prove the inextendibility of spherically symmetric weak null singularities as spherically symmetric Lorentzian manifolds with a continuous metric and Christoffel symbols in $L^{s}_{\text{loc}}$ for $s>1$. The result assumes a suitable blow-up condition on the derivative of the area-radius function transverse to the weak null singularity in combination with a rigidity result on continuous spherically symmetric extensions across null boundaries proven in [1]. In particular we show that these assumptions are satisfied by the Reissner-Nordström-Vaidya spacetime as well as by a class of spacetimes arising from small and generic spherically symmetric perturbations of subextremal Reissner-Nordström under the Einstein-Maxwell-scalar field system.

Dimension of partially ordered sets (posets for short) can be seen as a measure of how much space is needed to store posets. Refining the definition of dimension, or actually the local dimension, in this respect we obtain the notion of relative dimension. We discuss properties of relative dimension and we give bounds for relative dimension of some well-known families of posets.

In this paper, we prove that if $E$ is a pseudo-effective sheaf with vanishing first Chern class on a klt compact Kähler space $X$, then, after passing to a finite quasi-étale cover, the reflexive pullback of $E$ is locally free and flat. This extends the flatness criterion of Höring–Peternell, originally established for projective varieties, to the Kähler setting. The proof relies on two main ingredients, both of which are new even in the projective case. The first is a flatness theorem for stable sheaves: we show that a slope-stable pseudo-effective sheaf with vanishing first Chern class is Hermitian flat. This is obtained by combining Hermitian–Einstein theory with the subharmonicity properties of direct image sheaves. The second is a singular Kähler analogue of Simpson's flatness theorem for extensions of locally free Hermitian flat sheaves.

We study super-matrices $M_{m|n}(R)$ over the rank-two exterior algebra $R=K[\theta_1,\theta_2]/(\theta_1^2,\theta_2^2,\theta_1\theta_2+\theta_2\theta_1)$, the smallest supercommutative ring on which the Berezinian's odd-by-odd correction term $BD^{-1}C$ need not vanish. We recall that the supertrace satisfies graded cyclicity, $\mathrm{str}(XY)=(-1)^{|X||Y|}\mathrm{str}(YX)$, so $\mathrm{str}([A,B])=0$ for even $A,B$, and that an explicit $1|1$ matrix over $R$ has $BD^{-1}C\neq0$ surviving to second order in the odd generators. We then prove the extension anticipated in the $1|1$ note: a closed-form Berezinian formula for arbitrary even $X\in M_{m|n}(R)$, $m,n\geq1$ (Theorem 1), given through a single $m\times m$ matrix pairing $\Delta$ built from the odd blocks of $X$; and a fully general multiplicativity theorem $\mathrm{Ber}(XY)=\mathrm{Ber}(X)\mathrm{Ber}(Y)$ for even invertible $X,Y\in M_{m|n}(R)$ (Theorem 2), proved by an explicit trace-cyclicity cancellation. Both results specialize exactly to the known $1|1$ formulas when $m=n=1$, and we verify them on a worked $m=2,n=1$ example. We extend the framework from two to an arbitrary number $r\geq2$ of odd generators, replacing the scalar pairing $\Delta$ by a $\Lambda^2(V)$-valued pairing on the $r$-dimensional space $V$ of odd generators, and identify this pairing, in the scalar case, with a Plücker coordinate of the odd off-diagonal data. We close with a discussion of related work of Khudaverdian–Voronov, and a remark identifying $\Delta$ as a matrix-valued contraction against the natural alternating pairing on the space of odd generators.

We introduce a new method for studying mod-$\ell$ congruences between eigenforms through the modular representation theory of $\mathrm{PGL}_2(\mathbb{F}_p)$. When $p\equiv \pm 1 \pmod{\ell}$, we use this theory to construct and describe extra structures on spaces of modular forms with $\Gamma_0(p^2)$-level at $p$ and a fixed mod-$\ell$ Galois representation. The structural results we obtain can be viewed as a refinement of classical level-raising theorems since they not only allow us to prove the existence of congruences, but also to count the number of such congruences. Our methods work equally well in the residually irreducible and residually reducible cases, allowing us to prove several new instances of congruences between Eisenstein series and cuspforms (as well as independently rederiving classical results of Mazur and more recent results of Lang–Wake). Notably, our approach proves these results without computing constant terms of Eisenstein series, without Galois deformation theory and $R=\mathbb{T}$ theorems, and without the Jacquet–Langlands correspondence.

We present a simple algebraic proof of meromorphic non-integrability of a 'mysterious' case of a quadratic Hamiltonian studied by Sokolov and Wolf. The result is obtained by applying the Morales-Ramis method, based on differential Galois theory.

We prove real-rootedness for the Chow polynomials of the noncrossing partition lattices by transferring tieless parking functions to finite-alphabet Smirnov words and applying an interlacing-preserving transition of M.~Leander. We also derive a triangular recurrence for peaks and ties and identify the peakless-tieless descent polynomial as the Narayana polynomial. For the toric $g$-contributions of Ehrenborg–Hetyei–Readdy, we exhibit a fixed-row common interlacer and establish real-rootedness of all nonnegative row sums. Individual real-rootedness follows in particular; Q.~Xiao also proved it by a different differential recurrence. We also give a second proof of the individual statement, by finite Schur–Szegő convolution, that does not use the common interlacer. These results prove Conjecture~4.2 of Xiao and Conjecture~11.2 of Ehrenborg–Hetyei–Readdy, with consequences for weakly 123-avoiding parking functions. We also prove real-rootedness for the image-size polynomial on all parking functions and for the ascent and descent polynomials of four two-pattern-avoiding classes.

Let $P$ denote normalized nearest-neighbor averaging on $\mathbb{Z}^d$ and set $A=I-P$. We consider $\partial_tu+A^su=f(u),t>0, x\in\mathbb{Z}^d, 0<s<1.$ For a Fisher-KPP reaction with $a=f'(0)>0$, compactly supported data have logarithmic propagation rate $a/(d+2s)$. In the logistic case, each fixed level set lies between two constant multiples of $\exp(at/(d+2s))$; this localization remains valid for data with the critical algebraic tail. Nondecreasing front-like data in one dimension instead propagate at rate $a/(2s)$, and the resulting acceleration rules out planar traveling fronts of finite speed in every dimension. We also analyze the singular limit $s\uparrow1$. If $\tau_s=-\log(1-s)$, then throughout $at<\tau_s$ compactly supported solutions follow the Wulff shape of the local nearest-neighbor equation. After a constant multiple of $\tau_s$, their propagation is exponential. Half-space data exhibit the corresponding transition from the directional local speed to the exponent $a/(2s)$. The analysis rests on sharp bounds for the subordinated kernel, algebraic barriers, a compactness–Liouville argument, and a decomposition of the fractional walk into its nearest-neighbor and rare long-jump parts.

We study constructible $\mathbb{Z}[\tfrac{1}{p}]$-linear étale motives on parahoric Hecke stacks associated with quasi-split tamely ramified reductive groups. We prove a canonical equivalence between their equal-characteristic and Witt-vector incarnations that is monoidal with respect to convolution. This extends Bando's comparison, which concerns split reductive groups and constructible étale sheaves. The proof combines the study of a family of affine Grassmannians for Pappas–Zhu group schemes with the use of the motivic nearby cycles functor.

In this paper, we investigate models of percolation of words on $\mathbb{Z}_+$ with long-range connections. The underlying graph is oriented and constructed according to a sequence of non-negative ranges. In the first model, the sequence of ranges is considered random and, in the second model, deterministic and non-decreasing. In both cases, we establish a phase transition for the occurrence of percolation of all words simultaneously from the origin.

Inspired by the monotonicity formula of Alt, Caffarelli, and Friedman, we consider a natural variant of the ACF functional in positively curved 2-dimensional settings where we integrate over sublevel sets of Green's function rather than disks. We prove sharp almost-monotonicity formulas for our new functional in the case of convex planar domains (both with the pole on the boundary and in the interior) and in the setting of 2-dimensional complete manifolds with Euclidean volume growth and nonnegative Gaussian curvature. The tools include the Schwarz-Christoffel formula and Riesz decomposition using conformal coordinates. As a consequence, using quasiconformal estimates, we give a new Lipschitz bound in the manifold setting for minimizers of the two-phase free boundary problem of Alt, Caffarelli, and Friedman, with the constant depending on only the total integral curvature, in contrast to work of Teixeira and Zhang, which gives constants depending on pointwise bounds for the Riemann curvature tensor and its derivatives. Our methods also recover the Lipschitz bound up to a Neumann boundary of Gemmer, Moon, and Raynor in the planar convex domain case.

In this paper, we study the Calabi curvature operator on Kähler manifolds. First, we prove that if the Calabi curvature operator on Kähler manifolds satisfies $\frac{{n\left(n + 1\right)}}{2}$-positive (nonnegative), $\frac{{n + 1}}{2}$-positive (nonnegative), and $\left( n-1 \right)$-positive (nonnegative), then the scalar curvature, Ricci curvature, and orthogonal Ricci curvature are positive (nonnegative), respectively. Second, we show that any compact Kähler Einstein manifold satisfying the condition $$\lambda_1+\dots+\lambda_{\alpha}\ge -{\alpha}\theta(n,{\alpha})\bar\lambda,\; {\alpha}\le \frac{n}{2}$$ must have nonnegative constant holomorphic sectional curvature.

We construct an explicit dictionary between two local classifications in dimension two: normal forms of pseudo-Riemannian metrics with a quadratic integral of the geodesic flow and normal forms of $\operatorname{gl}$-regular Nijenhuis operators. The correspondence is obtained by an explicit change of coordinates constructed from the trace and determinant of the operator.

We develop novel duality methods for stochastic optimal control problems under two sources of randomness and non-Markovian random coefficients adapted to just one of them. The Hamilton-Jacobi-Bellman (HJB) equation is a second-order backward stochastic partial differential equation. Our duality theory provides an alternative description of the random value function in terms of a suitable pathwise optimal control problem parameterized by the realizations of one of the Brownian motions. This allows us to regain Markovianity using the theory of rough optimal control problems, leading to rough second order HJB equations that we solve in a suitable viscosity sense.

The purpose of this work is twofold. In the first part, we consider layered mixed matrices introduced by Murota, relate them to existing notions in combinatorial commutative algebra, and investigate the irreducibility of their determinants. Furthermore, for a layered mixed matrix in combinatorial canonical form, we determine the sparsity structure of its inverse. That is, we characterize which entries of the inverse are nonzero. In the second part, we establish for the first time a formal connection between these algebraic results and the theory of buffering structures for reaction networks developed by Mochizuki and Okada. We identify the lattice of buffering structures with the lattice of order ideals of the block poset of the combinatorial canonical form of the associated layered mixed matrix. This allows us to characterize the reducibility of the symbolic Jacobian determinant as a polynomial in the reaction-rate derivatives, as well as the nonzero sensitivity responses of species concentrations to reaction-rate perturbations.

We introduce an admissible trembling-hand refinement for stochastic mean field games formulated through relaxed controlled martingale problems. A relaxed MFG equilibrium is perfect if it is the common Wasserstein limit of two sequences of joint control-state laws. The first consists of full-support population perturbations admissible for the state laws they generate; the second consists of exact optimal responses to those perturbed state laws. This admissibility condition prevents the control and state coordinates of a perturbation from being varied independently. Under the standing continuity, growth, and coercivity assumptions, we prove existence without requiring compact controls or bounded coefficients. The proof combines a perturbed fixed-point argument in the compact-bounded case with truncation, uniform moment estimates, compactness, and stability of controlled martingale problems. A one-dimensional model with exactly two relaxed mean field game equilibria shows that the refinement is genuinely selective: only one equilibrium is trembling-hand perfect.

We develop a framework for simultaneous change-point inference of high-dimensional functional time series. The observations are modeled as temporally dependent vectors whose coordinates take values in possibly different separable Hilbert spaces, thereby covering a broad class of functional data. Heterogeneous mean changes may occur at coordinate-specific locations, and the contemporaneous dependence across coordinates is left unrestricted. Our procedure is based on coordinatewise cumulative-sum statistics and a residual block multiplier bootstrap that provides a common critical value for the global test and all coordinatewise decisions. We establish a nonasymptotic Gaussian approximation, quantitative nonasymptotic bounds for strong family-wise error control under arbitrary mixtures of changed and unchanged coordinates, covariance-adaptive detection guarantees, and simultaneous high-probability bounds for change-point localization. The bounds accommodate high-dimensional regimes in which the number of functional coordinates grows exponentially in a power of the sample size. We investigate finite-sample performance in simulations and illustrate the method using river discharge curves and high-frequency financial log returns.

We prove that a class of nonlocal nonlinearities on periodic Sobolev spaces can be expressed as a trace of the unique mild solution to a local dispersive problem consisting of a system of forced Schrödinger equations. The proof is based on a reformulation of the nonlocal nonlinearities as the resonant orbit average of a real-analytic function under the unitary group generated by the free Schrödinger operator $\mathrm{i} \partial_x^2$. We illustrate our result by providing an equivalent local reformulation for a recently obtained nonlocal amplitude equation that formally captures the dynamics of parabolic systems close to a conserved Hopf instability.