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作者：Ganatra, Sheel; Pardon, John; Shende, Vivek
摘要：The Nadler-Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms at infinity, given on the Floer side by Reeb trajectories (also known as wrapping) and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact....
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作者：Paolini, Gianluca; Shelah, Saharon
摘要：We prove that the Borel space of torsion -free abelian groups with domain omega is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a longstanding open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.
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作者：Regev, Oded; Stephens-Davidowitz, Noah
摘要：We prove a conjecture due to Dadush, showing that if L subset of R-n is a lattice such that det (L ') >= 1 for all sublattices L 'subset of L, then & sum;(e-pi t2 parallel to y parallel to 2 <= 3/2, )(y is an element of L)where t:=10(logn+2). From this we derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski's celebrated first theorem. We also derive a bound on the covering radius.
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作者：Anari, Nima; Liu, Kuikui; Gharan, Shayan Oveis; Vinzant, Cynthia
摘要：We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0 < q < 1. Consequently, we can sample random spanning forests in a graph and estimate the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has edge expansion at least 1. Our algorithm and proof...
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作者：Cai, Yuanqing; Friedberg, Solomon; Kaplan, Eyal
摘要：We study the generalized doubling method for pairs of representations of G x GLk where G is a symplectic group, split special orthogonal group or split general spin group. We analyze the poles of the local integrals and prove that the global completed L-function with a cuspidal representation of GLk twisted by a highly ramified Hecke character is entire. We obtain a new proof of the weak functorial transfer of cuspidal automorphic representations of G to the natural general linear group, which...
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作者：Maulik, Davesh; Shen, Junliang
摘要：We prove the P = W conjecture for GL(n) for all ranks n and curves of arbitrary genus g >= 2. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the perversity statement, we apply the vanishing cycles constructions in our earlier work to global Springer theory in the sense of Yun, and prove a parabolic support theorem.
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作者：Binyamini, Gal; Novikov, Dmitry; Zak, Benny
摘要：1.1. Sharply o-minimal structures. Our general results in this paper are stated in the context of sharply o-minimal structures admitting sharp derivatives. For a complete definition of these notions, see Section 2. Briefly, a sharply o-minimal structure (abbreviated #o-minimal) is a pair (S, S2), where S is an o-minimal expansion of the real field and we identify S with the collection of all definable sets in S; and S2 = {S2 ,D c S} ,DEN is filtration on S depending on the format and degree D....
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作者：Li, Penghui; Nadler, David; Yun, Zhiwei
摘要：For a complex reductive group, we construct a semi-orthogonal decomposition of the cocenter of the universal variant of its affine Hecke category. We use this to calculate the endomorphisms of a Whittaker object in the cocenter via a diagram organizing parabolic induction of character sheaves. Assuming a universal variant of Bezrukavnikov's spectral description of the affine Hecke category, we deduce a formula for the dg algebra of global functions on commuting stacks of complex reductive grou...
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作者：Lee, Yin Tat; Vempala, Santosh S.
摘要：We analyze the Poincar & eacute; and Log-Sob olev constants of logconcave densities in Rn. For the Poincar & eacute; constant, we give an improved estimate of O(root n) for any isotropic logconcave density. For the Log-Sob olev constant, we prove a bound of Omega(1/D), where D is the diameter of the support of the density, and show that this is asymptotically the best possible bound, resolving a question posed by Frieze and Kannan in 1997. These bounds have several interesting consequences. Im...
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作者：Dunn, Alexander; Radziwill, Maksym
摘要：Let W be a smooth test function with compact support in (0, oo). Conditional on the Generalized Riemann Hypothesis for Hecke L-functions over Q ( omega ), we prove that X p equivalent to 1 (mod 3) 1 ( X e 2 pi ix 3 /p ) ( p ) 2 root p WX x (mod p ) (2 pi)2/3 Z infinity ti W(x)x-1/6dx X 5 / 6 3 Gamma(23) logX, 0 as X-* oo and p runs over primes. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846 and confirms (conditionally on the G...
