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The many faces of the stochastic zeta function Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-08-18 Benedek Valkó, Bálint Virág
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
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Meromorphic $$L^2$$ L 2 functions on flat surfaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-08-11 Ian Frankel
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Functionals with extrema at reproducing kernels Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-07-12 Aleksei Kulikov
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.
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Random Walks with Bounded First Moment on Finite-volume Spaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-07-05 Timothée Bénard, Nicolas de Saxcé
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Lord Rayleigh’s Conjecture for Vibrating Clamped Plates in Positively Curved Spaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-07-01 Alexandru Kristály
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A random cover of a compact hyperbolic surface has relative spectral gap $$\frac{3}{16}-\varepsilon $$ 3 16 - ε Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-05-17 Michael Magee, Frédéric Naud, Doron Puder
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Analytic Langlands correspondence for $$PGL_2$$ P G L 2 on $${\mathbb {P}}^1$$ P 1 with parabolic structures over local fields Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-05-18 Pavel Etingof, Edward Frenkel, David Kazhdan
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
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Lamplighters and the bounded cohomology of Thompson’s group Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-05-11 Nicolas Monod
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Dimers, networks, and cluster integrable systems Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-05-11 Anton Izosimov
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Ricci Solitons, Conical Singularities, and Nonuniqueness Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-04-22 Sigurd B. Angenent, Dan Knopf
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Steenrod Pseudocycles, Lifted Cobordisms, and Solomon’s Relations for Welschinger Invariants Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-04-15 Xujia Chen
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Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-04-10 Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, De-Qi Zhang
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 \((g^m)^* \circlearrowleft H^2(X,{{\mathbb {C}}})\)
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Joints of Varieties Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-04-09 Jonathan Tidor, Hung-Hsun Hans Yu, Yufei Zhao
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
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Symplectic mapping class groups of K3 surfaces and Seiberg–Witten invariants Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-03-31 Gleb Smirnov
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Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-03-28 Gang Liu, Gábor Székelyhidi
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
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Ramanujan Complexes and Golden Gates in PU(3) Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-03-26 Shai Evra, Ori Parzanchevski
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)
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Examples of Ricci limit spaces with non-integer Hausdorff dimension Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-03-22 Jiayin Pan, Guofang Wei
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.
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Random hyperbolic surfaces of large genus have first eigenvalues greater than $$\frac{3}{16}-\epsilon $$ 3 16 - ϵ Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-03-17 Yunhui Wu, Yuhao Xue
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
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The Patterson–Sullivan Reconstruction of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-03-15 Alexander I. Bufetov, Yanqi Qiu
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
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Publisher Correction to: Point-like bounding chains in open Gromov–Witten theory Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-02-01 Jake P. Solomon, Sara B. Tukachinsky
Due to a processing error, the affiliations of authors were published incorrectly.
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The exact value of Hausdorff dimension of escaping sets of class $${\mathcal {B}}$$ B meromorphic functions Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-01-25 Volker Mayer, Mariusz Urbański
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
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Non-uniqueness of minimal surfaces in a product of closed Riemann surfaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-01-17 Vladimir Marković
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.
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Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-01-07 Wencai Liu
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
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On arithmetic sums of Ahlfors-regular sets Geom. Funct. Anal. (IF 1.926) Pub Date : 2022-01-05 Tuomas Orponen
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 \(\dim _{\mathrm {H}}E = 0\).
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Volume and macroscopic scalar curvature Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-12-28 Sabine Braun, Roman Sauer
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
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Idempotent Fourier multipliers acting contractively on $$H^p$$ H p spaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-12-27 Brevig, Ole Fredrik, Ortega-Cerdà, Joaquim, Seip, Kristian
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 \(p=2(n+1)\)
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Point-like bounding chains in open Gromov–Witten theory Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-12-22 Solomon, Jake P., Tukachinsky, Sara B.
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
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Ergodic theory of affine isometric actions on Hilbert spaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-12-15 Arano, Yuki, Isono, Yusuke, Marrakchi, Amine
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
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Isometry group of Lorentz manifolds: A coarse perspective Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-11-30 Frances, Charles
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
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Singularity of discrete random matrices Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-10-22 Jain, Vishesh, Sah, Ashwin, Sawhney, Mehtaab
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
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The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-10-20 Logunov, A., Malinnikova, E., Nadirashvili, N., Nazarov, F.
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
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Uniformization of compact complex manifolds by Anosov homomorphisms Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-09-14 Dumas, David, Sanders, Andrew
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
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A proof of Ringel’s conjecture Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-09-02 Montgomery, R., Pokrovskiy, A., Sudakov, B.
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
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A numerical criterion for generalised Monge-Ampère equations on projective manifolds Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-09-01 Datar, Ved V., Pingali, Vamsi Pritham
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
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Sweepouts of closed Riemannian manifolds Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-08-31 Nabutovsky, Alexander, Rotman, Regina, Sabourau, Stéphane
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
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Monodromy groups of Kloosterman and hypergeometric sheaves Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-08-29 Katz, Nicholas M., Tiep, Pham Huu
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
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Conformal upper bounds for the volume spectrum Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-08-24 Wang, Zhichao
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.
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Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-08-13 Girouard, Alexandre, Karpukhin, Mikhail, Lagacé, Jean
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
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Generic scarring for minimal hypersurfaces along stable hypersurfaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-07-02 Antoine Song, Xin Zhou
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
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Local uniformity through larger scales Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-06-30 Miguel N. Walsh
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.
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Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-06-18 S. Yu. Orevkov
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
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Building manifolds from quantum codes Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-06-18 Michael Freedman, Matthew Hastings
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
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Deformations of Totally Geodesic Foliations and Minimal Surfaces in Negatively Curved 3-Manifolds Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-06-02 Ben Lowe
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
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Entangleability of cones Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-05-15 Guillaume Aubrun, Ludovico Lami, Carlos Palazuelos, Martin Plávala
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
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Non-asymptotic Results for Singular Values of Gaussian Matrix Products Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-05-08 Boris Hanin, Grigoris Paouris
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
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Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-05-08 Steve Hofmann, José María Martell, Svitlana Mayboroda, Tatiana Toro, Zihui Zhao
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
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Flows on measurable spaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-05-08 László Lovász
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
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A pair correlation problem, and counting lattice points with the zeta function Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-05-08 Christoph Aistleitner, Daniel El-Baz, Marc Munsch
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
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The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-04-10 Sherry Gong, Jianchao Wu, Guoliang Yu
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
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Non-displaceable Lagrangian links in four-manifolds Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-04-08 Cheuk Yu Mak, Ivan Smith
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
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Short geodesic loops and $$L^p$$ L p norms of eigenfunctions on large genus random surfaces Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-04-05 Clifford Gilmore, Etienne Le Masson, Tuomas Sahlsten, Joe Thomas
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
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Simultaneous Small Fractional Parts of Polynomials Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-04-01 James Maynard
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
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On the variance of squarefree integers in short intervals and arithmetic progressions Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-03-31 Ofir Gorodetsky, Kaisa Matomäki, Maksym Radziwiłł, Brad Rodgers
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
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An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-03-24 Yuansi Chen
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
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$$\mathbf {Bad}\left( {\mathbf {w}}\right) $$ Bad w is hyperplane absolute winning Geom. Funct. Anal. (IF 1.926) Pub Date : 2021-03-13 Victor Beresnevich, Erez Nesharim, Lei Yang
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
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A landing theorem for entire functions with bounded post-singular sets Geom. Funct. Anal. (IF 1.926) Pub Date : 2020-11-20 Anna Miriam Benini, Lasse Rempe
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
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Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows Geom. Funct. Anal. (IF 1.926) Pub Date : 2020-11-20 Karl-Theodor Sturm
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 \(\psi \in \mathrm {Lip}_b(X)\)
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Classical Theta Lifts for Higher Metaplectic Covering Groups Geom. Funct. Anal. (IF 1.926) Pub Date : 2020-11-18 Solomon Friedberg, David Ginzburg
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
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Partial associativity and rough approximate groups Geom. Funct. Anal. (IF 1.926) Pub Date : 2020-11-08 W. T. Gowers, J. Long
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
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Cohomological obstructions to lifting properties for full C $$^*$$ ∗ -algebras of property (T) groups Geom. Funct. Anal. (IF 1.926) Pub Date : 2020-10-26 Adrian Ioana, Pieter Spaas, Matthew Wiersma
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)