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作者：Allen, Patrick B.; Calegari, Frank; Caraiani, Ana; Gee, Toby; Helm, David; Le Hung, Bao V.; Newton, James; Scholze, Peter; Taylor, Richard; Thorne, Jack A.
摘要：Let F be a CM number field. We prove modularity lifting theorems for regular n-dimensional Galois representations over F without any self-duality condition. We deduce that all elliptic curves E over F are potentially mod-ular, and furthermore satisfy the Sato-Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for GL2(AF).
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作者：Edelen, Nick; Spolaor, Luca
摘要：Hardt-Simon proved that every area-minimizing hypercone C having only an isolated singularity fits into a foliation of Rn+1 by smooth, areaminimizing hypersurfaces asymptotic to C. In this paper we prove that if a stationary integral n-varifold M in the unit ball B-1 subset of Rn+1 lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone C-3,C-3), then spt M boolean AND B-1/2 is a C-1,C- alpha perturbation of either the cone itself, or some leaf of its associated f...
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作者：Dittmann, Philip; Pop, Florian
摘要：We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class, there is a first-order sentence that characterizes this field within the class up to isomorphism. Our solution is conditional on resolution of singularities in characteristic two and unconditional in all other characteristics.
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作者：Conde-Alonso, Jose M.; Gonzalez-Perez, Adrian M.; Parcet, Javier; Tablate, Eduardo
摘要：We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers S-M on Schatten p-classes which solves a conjecture proposed by Mikael de la Salle. Given 1 < p < infinity, a simple form of our main result for R-n x R-n matrices reads as follows: ||S-M : S-p -> S-p||(cb) less than or similar to p(2)/p-1 Sigma(|gamma|less than or similar to[n/2]+1)|||x y|(|gamma|) {|partial derivative M-gamma(x)(x, y) | + |partial derivative M-gamma(y)(x , y)|}||(proportional to). ...
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作者：Schumacher, Georg; Tsuji, Hajime
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作者：Aggarwal, Amol
摘要：In this paper we consider uniformly random lozenge tilings of arbitrary domains approximating (after suitable normalization) a closed, simply-connected subset of R-2 with piecewise smooth, simple boundary. We show that the local statistics of this model around any point in the liquid region of its limit shape are given by the infinite-volume, translation-invariant, extremal Gibbs measure of the appropriate slope, thereby confirming a prediction of Cohn-Kenyon-Propp from 2001 in the case of loz...
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作者：Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymond, Laure; Simonella, Sergio
摘要：We present a mathematical theory of dynamical fluctuations for the hard sphere gas in the Boltzmann-Grad limit. We prove that (1) fluctuations of the empirical measure from the solution of the Boltzmann equation, scaled with the square root of the average number of particles, converge to a Gaussian process driven by the fluctuating Boltzmann equation, as predicted by Spohn; (2) large deviations are exponentially small in the average number of particles and are characterized, under regularity a...
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作者：Chifan, Ionut; Ioana, Adrian; Osin, Denis; Sun, Bin
摘要：We introduce a new class of groups called wreath-like products. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan's property (T). In this paper, we prove that any group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group von Neumann algebra L(G) remembers the isomorphism class of G. This allows us to ...
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作者：Browning, Tim; Le Boudec, Pierre; Sawin, Will
摘要：It is known that the Brauer-Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least 3 over any number field. Moreover, for such varieties it follows from a general con-jecture of Colliot-The ' le`ne that the Brauer-Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hy-persurfaces of fixed degree and dimension by height, we p...
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作者：Song, Antoine
摘要：Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.
