We introduce a framework to study the random entire function \(\zeta _\beta \) whose zeros are given by the Sine\(_\beta \) process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power series representation built from Brownian motion. We study related distributions using stochastic differential equations. Our function is a uniform limit of

We show that certain monotone functionals on the Hardy spaces and convex functionals on the Bergman spaces are maximized at the normalized reproducing kernels among the functions of norm 1, thus proving the contractivity conjecture of Pavlović and of Brevig, Ortega-Cerdà, Seip and Zhao and the Wehrl-type entropy conjecture for the SU(1, 1) group of Lieb and Solovej, respectively.

We continue to develop the analytic Langlands program for curves over local fields initiated in our earlier papers, following a suggestion of Langlands and a work of Teschner. Namely, we study the Hecke operators which we introduced in those papers in the case of a projective line with parabolic structures at finitely many points for the group \(PGL_2\). We establish most of our conjectures in this

We study zero entropy automorphisms of a compact Kähler manifold X. Our goal is to bring to light some new structures of the action on the cohomology of X, in terms of the so-called dynamical filtrations on \(H^{1,1}(X,{{\mathbb {R}}})\). Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations \((g^m)^* \circlearrowleft H^2(X,{{\mathbb {C}}})\)

We generalize the Guth–Katz joints theorem from lines to varieties. A special case says that N planes (2-flats) in 6 dimensions (over any field) have \(O(N^{3/2})\) joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of N planes is replaced by a set of 2-dimensional algebraic varieties of total

We show that non-collapsed Gromov–Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson–Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show

In a seminal series of papers from the 80’s, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for \(GL_{2}\) (Deligne’s theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with \(SL_{2}({\mathbb {Q}}_{p})\). As a result, they obtained explicit Ramanujan Cayley graphs from \(PSL_{2}\left( {\mathbb {F}}_{p}\right)

We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding [CC00a, Page 15] about collapsing Ricci limit spaces.

Let \(\mathcal {M}_g\) be the moduli space of hyperbolic surfaces of genus g endowed with the Weil–Petersson metric. In this paper, we show that for any \(\epsilon >0\), as genus g goes to infinity, a generic surface \(X\in \mathcal {M}_g\) satisfies that the first eigenvalue \(\lambda _1(X)>\frac{3}{16}-\epsilon \). As an application, we also show that a generic surface \(X\in \mathcal {M}_g\) satisfies

The Patterson–Sullivan reconstruction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball \({\mathbb {B}}_d\) in \({\mathbb {C}}^d\). For supercritical weighted Bergman spaces, the reconstruction is uniform when the functions range over the unit ball of the weighted

Due to a processing error, the affiliations of authors were published incorrectly.

We consider the subclass of class \({\mathcal {B}}\) consisting of meromorphic functions \(f:{{\mathbb {C}}}\rightarrow \widehat{{{\mathbb {C}}}} \) for which infinity is not an asymptotic value and whose all poles have orders uniformly bounded from above. This class was introduced in Bergweiler and Kotus (Trans Am Math Soc 364(10):5369–5394, 2012) and the Hausdorff dimension \(\text {HD}({\mathcal

We show that for every large enough \(\mathbf {g}\) there exists a Fuchsian representation \(\rho : \pi _1(\Sigma _\mathbf{{g}}) \rightarrow \prod _{i=1}^{3} \, {\mathbf {PSL}}(2,{\mathbb {R}} )\) which yields multiple minimal surfaces in the corresponding product of closed Riemann surfaces.

Let \(H_0\) be a discrete periodic Schrödinger operator on \(\ell ^2(\mathbb {Z}^d)\): $$\begin{aligned} H_0=-\Delta +V, \end{aligned}$$ where \(\Delta \) is the discrete Laplacian and \(V:\mathbb {Z}^d\rightarrow \mathbb {C}\) is periodic. We prove that for any \(d\ge 3\), the Fermi variety at every energy level is irreducible (modulo periodicity). For \(d=2\), we prove that the Fermi variety at every

Let \(A,B \subset \mathbb {R}\) be closed Ahlfors-regular sets with dimensions \(\dim _{\mathrm {H}}A =: \alpha \) and \(\dim _{\mathrm {H}}B =: \beta \). I prove that $$\begin{aligned} \dim _{\mathrm {H}}[A + \theta B] \ge \alpha + \beta \cdot \tfrac{1 - \alpha }{2 - \alpha } \end{aligned}$$ for all \(\theta \in \mathbb {R}{\setminus } E\), where \(\dim _{\mathrm {H}}E = 0\).

We prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of \(\ell ^2\)-Betti numbers of aspherical Riemannian manifolds in the presence of a lower

We describe the idempotent Fourier multipliers that act contractively on \(H^p\) spaces of the d-dimensional torus \(\mathbb {T}^d\) for \(d\ge 1\) and \(1\le p \le \infty \). When p is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on \(L^p\) spaces, which in turn can be described by suitably combining results of Rudin and Andô. When \(p=2(n+1)\)

We present a solution to the problem of defining genus zero open Gromov–Witten invariants with boundary constraints for a Lagrangian submanifold of arbitrary dimension. Previously, such invariants were known only in dimensions 2 and 3 from the work of Welschinger. Our approach does not require the Lagrangian to be fixed by an anti-symplectic involution, but can use such an involution, if present, to

The classical Gaussian functor associates to every orthogonal representation of a locally compact group G a probability measure preserving action of G called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of G on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle

We prove a structure theorem for the isometry group \({\text {Iso}}(M,g)\) of a compact Lorentz manifold, under the assumption that a closed subgroup has exponential growth. We don’t assume anything about the identity component of \({\text {Iso}}(M,g)\), so that our results apply for discrete isometry groups. We infer a full classification of lattices that can act isometrically on compact Lorentz manifolds

Let \(\xi \) be a non-constant real-valued random variable with finite support and let \(M_{n}(\xi )\) denote an \(n\times n\) random matrix with entries that are independent copies of \(\xi \). For \(\xi \) which is not uniform on its support, we show that $$\begin{aligned} {\mathbb {P}}[M_{n}(\xi )\text { is singular}]&= {\mathbb {P}}[\text {zero row or column}] \\ {}&\quad +(1+o_n(1)){\mathbb {P}}[\text

Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^n\) with \(C^{1}\) boundary and let \(u_\lambda \) be a Dirichlet Laplace eigenfunction in \(\Omega \) with eigenvalue \(\lambda \). We show that the \((n-1)\)-dimensional Hausdorff measure of the zero set of \(u_\lambda \) does not exceed \(C(\Omega )\sqrt{\lambda }\). This result is new even for the case of domains with \(C^\infty \)-smooth boundary

We study uniformization problems for compact manifolds that arise as quotients of domains in complex flag varieties by images of Anosov homomorphisms. We focus on Anosov homomorphisms with “small” limit sets, as measured by the Riemannian Hausdorff codimension in the flag variety. Under such a codimension hypothesis, we show that all first-order deformations of complex structure on the associated compact

A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs \(2n+1\) times into the complete graph \(K_{2n+1}\). In this

We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the J-equation

We show that for every closed Riemannian manifold there exists a continuous family of 1-cycles (defined as finite collections of disjoint closed curves) parametrized by a sphere and sweeping out the whole manifold so that the lengths of all connected closed curves are bounded in terms of the volume (or the diameter) and the dimension n of the manifold, when \(n \ge 3\). An alternative form of this

We study the possible structures of monodromy groups of Kloosterman and hypergeometric sheaves on \({{\mathbb {G}}}_m\) in characteristic p. We show that most such sheaves satisfy a certain condition \(\mathrm {(\mathbf{S+})}\), which has very strong consequences on their monodromy groups. We also classify the finite, almost quasisimple, groups that can occur as monodromy groups of Kloosterman and

In this paper, we prove upper bounds for the volume spectrum of a Riemannian manifold that depend only on the volume, dimension and a conformal invariant.

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual

Let \(M^{n+1}\) be a closed manifold of dimension \(3\le n+1\le 7\). We show that for a \(C^\infty \)-generic metric g on M, to any connected, closed, embedded, 2-sided, stable, minimal hypersurface \(S\subset (M,g)\) corresponds a sequence of closed, embedded, minimal hypersurfaces \(\{\Sigma _k\}\) scarring along S, in the sense that the area and Morse index of \(\Sigma _k\) both diverge to infinity

By associating frequencies to larger scales, we provide a simpler way to derive local uniformity of multiplicative functions on average from the results of Matomäki-Radziwiłł.

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in \({\mathbb {RP}}^2\) of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in \({\mathbb {CP}}^2\) of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented

We give a procedure for “reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over \({\mathbb {Z}}\). Applying this procedure to chain complexes obtained by “lifting" recently developed quantum codes, which correspond to chain complexes over \({\mathbb {Z}}_2\), we construct the first examples of power law \({\mathbb {Z}}_2\) systolic

Let \(g_t\) be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold M starting at the hyperbolic metric. We construct foliations of the Grassmann bundle \(Gr_2(M)\) of tangent 2-planes whose leaves are (lifts of) minimal surfaces in \((M,g_t)\). These foliations are deformations of the foliation of \(Gr_2(M)\) by (lifts of) totally geodesic planes projected down

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones \({\mathcal {C}}_1\), \({\mathcal {C}}_2\), their minimal tensor product is the cone generated by products of the form \(x_1\otimes x_2\), where

This article provides a non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where N, the number of terms in the product, is large and n, the size of the matrices, may be large or small and may depend on N. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate for convergence of the empirical measure of the

The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying

The theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that one of the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form \((a_n \alpha )_{n \ge 1}\) has been pioneered by Rudnick, Sarnak and Zaharescu. Here \(\alpha \) is a real parameter, and \((a_n)_{n \ge 1}\) is an integer sequence, often

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard

Let \(\omega \) denote an area form on \(S^2\). Consider the closed symplectic 4-manifold \(M=(S^2\times S^2, A\omega \oplus a \omega )\) with \(0

We give upper bounds for \(L^p\) norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus \(g \rightarrow +\infty \), we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have with high probability at most one such loop of length

Let \(f_1,\dots ,f_k\in \mathbb {R}[X]\) be polynomials of degree at most d with \(f_1(0)=\dots =f_k(0)=0\). We show that there is an \(n

We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length \(H < x^{6/11 - \varepsilon }\) and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with \(q > x^{5/11 + \varepsilon }\). On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges

We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency \(d^{-o_d(1)}\). When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency \(d^{-1/4}\). Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many

In 1998 Kleinbock conjectured that any set of weighted badly approximable \(d\times n\) real matrices is a winning subset in the sense of Schmidt’s game. In this paper we prove this conjecture in full for vectors in \({\mathbb {R}}^d\) in arbitrary dimensions by showing that the corresponding set of weighted badly approximable vectors is hyperplane absolute winning. The proof uses the Cantor potential

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue

We will study metric measure spaces \((X,\mathsf{d},{\mathfrak {m}})\) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds \(\mathsf{BE}_1(\kappa ,\infty )\) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary \(\psi \in \mathrm {Lip}_b(X)\)

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous

Suppose that a binary operation \(\circ \) on a finite set X is injective in each variable separately and also associative. It is easy to prove that \((X,\circ )\) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples \((x,y,z)\in X^3\) satisfy the equation \(x\circ (y\circ z)=(x\circ y)\circ z\). Other results in additive combinatorics would

We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C\(^*\)-algebras of countable groups with (relative) property (T). We derive that the full C\(^*\)-algebras of the groups \(\mathbb {Z}^2\times \text {SL}_2({\mathbb {Z}})\) and \(\text {SL}_n({\mathbb {Z}})\), for \(n\ge 3\), do not have the local lifting property (LLP)