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作者：Manolescu, Ciprian; Ozsvath, Peter s.; Thurston, Dylan p.
作者单位：Stanford University; Princeton University; Indiana University System; Indiana University Bloomington
摘要：We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2Z). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsv & aacute;th-Szab & oacute; mixed invariants of closed four-manifolds, also in terms of grid diagrams.
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作者：An, Xinliang
作者单位：National University of Singapore
摘要：Employing the Einstein-scalar field system, we demonstrate an approach for proving high co-dimensional nonlinear instability of naked-singularity solutions as constructed by Christodoulou in 1994. We further investigate the censorship of Christodoulou's naked singularity and show that a tiny anisotropic perturbation arising from the outgoing characteristic initial data would lead to the emergence of an anisotropic apparent horizon, which covers and censors the naked singularity. Our approach a...
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作者：Bhargava, Manjul
作者单位：Princeton University
摘要：Of the (2H+1)(n) monic integer polynomials f(x)=x(n)+a1x(n-1)+...+a(n) with max{|a(1)|,...,|a(n)|} <= H, how many have associated Galois group that is not the full symmetric group S-n?There are clearly >> Hn-1 such polynomials, as may be obtained by setting a(n)=0. In 1936, van der Waerden conjectured that O(Hn-1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n <= 4, due to work of van der Waerden and Chow...
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作者：Poels, Anthony
作者单位：Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Jean Monnet; Universite Lyon 1; Ecole Centrale de Lyon; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：In his seminal 1961 paper, Wirsing studied how well a given transcendental real number xi can be approximated by algebraic numbers alpha of degree at most n for a given positive integer n , in terms of the so-called naive height H(alpha) of alpha . He showed that the supremum omega & lowast;n(xi) of all omega for which infinitely many such alpha have |xi- alpha| <= H(alpha)-omega-1 is at least (n + 1)/2. He also asked if we could even have omega & lowast;n(xi) >= nas it is generally expected. ...
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作者：Kisin, Mark; Zhou, Bong
作者单位：Harvard University; University of Cambridge
摘要：Let A be an abelian variety over a number field E subset of C, and let G denote the Mumford-Tate group of A. After replacing E by a finite extension, the action of the absolute Galois group Gal(E/E) on the & ell;-adic cohomology H1et(AE, Q & ell;) factors through G(Q & ell;). We show that for v an odd prime of E where A has good reduction, the conjugacy class of Frobenius Frobvin G(Q & ell;) is independent of & ell;. Along the way, we prove that under certain hypotheses, every point in the mu ...
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作者：Cohen, Alex
作者单位：Massachusetts Institute of Technology (MIT)
摘要：We prove that if a fractal set in Rd avoids lines in a certain quantitative sense, which we call line porosity, then it has a fractal uncertainty principle. The main ingredient is a new higher-dimensional Beurling-Malliavin multiplier theorem.
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作者：Leslie, Spencer
作者单位：Boston College
摘要：We prove the endoscopic fundamental lemma for the Lie algebra of the symmetric variety U(2n)/U(n) x U(n), where U(n) denotes a unitary group of rank n. This is the first major step in the stabilization of the relative trace formula associated to the U(n) x U(n)-periods of automorphic forms on U(2n).
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作者：Lawrence, Brian; Sawin, Will
作者单位：Princeton University
摘要：Faltings proved that there are finitely many abelian varieties of genus g over a number field K, with good reduction outside a finite set of primes S. Fixing one of these abelian varieties A, we prove that there are finitely many smooth hypersurfaces in A, with good reduction outside S, representing a given ample class in the Neron-Severi group of A, up to translation, as long as the dimension of A is at least four. Our approach builds on the approach of Lawrence and Venkatesh, which studies p...
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作者：Kato, Syu
作者单位：Kyoto University
摘要：We explain that the Pontryagin product structure on the equivariant K-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. 146 (2010)] coincides with the tensor structure on the equivariant K-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke Math. 169 (2020)]. Then, we construct an explicit isomorphism between the equivariant K-group of a semi-infinite flag manifold and a suitably localized equivariant quantum K-group of the correspondi...
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作者：Eberle, Simon; Figalli, Alessio; Weiss, Georg s.
作者单位：Basque Center for Applied Mathematics (BCAM); Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Duisburg Essen
摘要：The characterization of global solutions to the obstacle problems in RN, or equivalently of null quadrature domains, has been studied for more than 90 years. In this paper, we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.
