Abstract: We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=\xi_0+\xi_1 x+\cdots+{1\over\sqrt{n!}}\xi_n x^n$, where $\xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real

Abstract: Let $F$ be a number field and let $\Bbb{A}_F$ be its ring of adeles. Let $B$ be a quaternion algebra over $F$ and let $\nu:B\to F$ be the reduced norm. Consider the reductive monoid $M$ over $F$ whose points in an $F$-algebra $R$ are given by $$ M(R):=\big\{\big(\gamma_1,\gamma_2\big)\in\big(B\otimes_F R\big)^2:\nu\big(\gamma_1\big)=\nu\big(\gamma_2\big)\big\}. $$ Motivated by an influential

Abstract: The primal cohomology $\Bbb{K}_{\Bbb{Q}}$ of the theta divisor $\Theta$ of a principally polarized abelian fivefold (ppav) is the direct sum of its invariant and anti-invariant parts $\Bbb{K}_{\Bbb{Q}}^{+1}$, resp. $\Bbb{K}_{\Bbb{Q}}^{-1}$ under the action of $-1$. For smooth $\Theta$, these have dimension $6$ and $72$ respectively. We show that $\Bbb{K}_{\Bbb{Q}}^{+1}$ consists of Hodge

Abstract: We consider the nonlinear heat equation $u_t-\Delta u=|u|^\alpha u$ on $\Bbb{R}^N$, where $\alpha>0$ and $N\ge 1$. We prove that in the range $0<\alpha<{4\over {N-2}}$, for every $\mu>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0(x)=\mu|x|^{-{2\over\alpha}}$. The construction is based on the analysis of the related inverted

Abstract: In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi,0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(\phi,0)$. It

Abstract: We consider a rigidity problem for the spectral gap of the Laplacian on an ${\rm RCD}(K,\infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng-Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional

Abstract: Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline{x})$ over a finite field of characteristic $p$ defines an Artin-Schreier-Witt tower of varieties whose Galois group is isomorphic to $\Bbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a nontrivial finite character of $\Bbb{Z}_p$ and a generic polynomial whose convex hull

Abstract: We give a simple sufficient condition for a Verdier quotient $\scr{T}/\scr{S}$ of a triangulated category $\scr{T}$ by a thick subcategory $\scr{S}$ to be realized inside of $\scr{T}$ as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot-Guo-Keller.

abstract: We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\scr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.

abstract: We present new free-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of one-parameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere

abstract: We establish an improvement of flatness result for critical points of Ginzburg-Landau energies with long-range interactions. It applies in particular to solutions of $(-\Delta)^{s/2}u=u-u^3$ in $\Bbb{R}^n$ with $s\in(0,1)$. As a corollary, we establish that solutions with asymptotically flat level sets are $1$D and prove the analogue of the De Giorgi conjecture (in the setting of minimizers)

abstract: S. Seto, L. Wang, and G. Wei proved that the gap between the first two Dirichlet eigenvalues of a convex domain in the unit sphere is at least as large as that for an associated operator on an interval with the same diameter, provided that the domain has the diameter at most $\pi/2$. In this paper, we extend Seto-Wang-Wei's result to convex domains in the unit sphere with diameter less than

abstract: We extend work of Denef and Sperber and also Cluckers regarding a conjecture of Igusa in the two dimensional setting by no longer requiring the polynomial to be nondegenerate with respect to its Newton diagram. More precisely we establish sharp, uniform bounds for complete exponential sums and the number of polynomial congruences for general quasi-homogeneous polynomials in two variables

abstract: Let $K$ be a discretely valued field with Henselian valuation ring and separably closed (but not necessarily perfect) residue field of characteristic $p$, $H$ a connected reductive $K$-group, and $\Theta$ a finite group of automorphisms of $H$. We assume that $p$ does not divide the order of $\Theta$ and Bruhat-Tits theory is available for $H$ over $K$ with $\scr{B}(H/K)$ the Bruhat-Tits

abstract: In this paper, we prove that there are no solutions for the curvature equation $$ \Delta u+e^u=8\pi n\delta_0\ {\rm on}\ E_{\tau},\quad n\in\Bbb{N}, $$ where $E_{\tau}$ is a flat rectangular torus and $\delta_0$ is the Dirac measure at the lattice points. This confirms a conjecture of Lin and Wang and also improves a result of Eremenko and Gabrielov. The nonexistence is a delicate problem

abstract: We establish quantitative estimates for sampling (dominating) sets in model spaces associated with meromorphic inner functions, i.e., those corresponding to de Branges spaces. Our results encompass the Logvinenko-Sereda-Panejah (LSP) Theorem including Kovrijkine's optimal sampling constants for Paley-Wiener spaces. It also extends Dyakonov's LSP theorem for model spaces associated with bounded