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作者：Hermon, Jonathan; Hutchcroft, Tom
作者单位：University of British Columbia; University of Cambridge
摘要：Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and p > p(c)(G) then there exists a positive constant c(p) such that P-p(n <= vertical bar K vertical bar < infinity) <= e(-cpn) for every n >= 1, where K is the cluster of the origin. We deduce the following two corollaries: 1. Every infinite cluster in supercritical percolation on a transitive non-amenable graph has anchored expansion almost surely. This ans...
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作者：Ancona, Giuseppe
作者单位：Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
摘要：Let A be an abelian fourfold in characteristic p. We prove the standard conjecture of Hodge type for A, namely that the intersection product Z(num)(2)(A)(Q) x Z(num)(2) (A)(Q) -> Q is of signature (rho(2) - rho(1) + 1; rho(1) - 1), with rho(n) = dim Z(num)(n)(A)(Q). (Equivalently, it is positive definite when restricted to primitive classes for any choice of the polarization.) The approach consists in reformulating this question into a p-adic problem and then using p-adic Hodge theory to solve...
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作者：Bruinier, Jan Hendrik; Ehlen, Stephan; Yang, Tonghai
作者单位：Technical University of Darmstadt; University of Cologne; University of Wisconsin System; University of Wisconsin Madison
摘要：Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function G(s) (z(1), z(2)) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable z(1) over all CM points of a fixed discriminant d(1) (twisted by a genus character), and allow in the second variable the evaluation a...
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作者：De Philippis, Guido; Spolaor, Luca; Velichkov, Bozhidar
作者单位：New York University; University of California System; University of California San Diego; University of Pisa
摘要：We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.
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作者：Looper, Nicole R.
作者单位：Brown University
摘要：We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the abc-conjecture. For unicritical polynomials of degree at least five, we require only the standard abc-conjecture.
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作者：Codogni, Giulio; Patakfalvi, Zsolt
作者单位：University of Rome Tor Vergata; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
摘要：The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary posi...
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作者：Hafner, Dietrich; Hintz, Peter; Vasy, Andras
作者单位：Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Massachusetts Institute of Technology (MIT); Stanford University
摘要：We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal ...
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作者：Aggarwal, Amol
作者单位：Columbia University
摘要：In this paper we analyze the large genus asymptotics for intersection numbers between psi-classes, also called correlators, on the moduli space of stable curves. Our proofs proceed through a combinatorial analysis of the recursive relations (Virasoro constraints) that uniquely determine these correlators, together with a comparison between the coefficients in these relations with the jump probabilities of a certain asymmetric simple random walk. As an application of this result, we provide the...
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作者：Bezrukavnikov, Roman; Vilonen, Kari
作者单位：Massachusetts Institute of Technology (MIT); University of Melbourne; University of Helsinki
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作者：Zhuang, Ziquan
作者单位：Massachusetts Institute of Technology (MIT)
摘要：We give an algebraic proof of the equivalence of equivariant K-semistability (resp. equivariant K-polystability) with geometric K-semistability (resp. geometric K-polystability). Along the way we also prove the existence and uniqueness of minimal optimal destabilizing centers on K-unstable log Fano pairs.
