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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：Rodnianski, Igor; Shlapentokh-Rothman, Yakov
摘要：In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in 3 + 1 dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation.Prior to this work, the only known examples of naked singularities wer...
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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：Guo, Bin; Phong, Duong H.; Tong, Freid
摘要：A PDE proof is provided for the sharp L & INFIN; estimates for the complex Monge-Ampe`re equation that had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics. It extends to more general fully non-linear equations satisfying a structural condition, and it also gives estimates of Trudinger type.
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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：D'Addezio, Marco
摘要：In this article we prove Crew's parabolicity conjecture of F-iso crystals. For this purpose, we introduce and study the notion of & DAG;-hull of a sub-F-iso-crystal. On the way, we prove a new Lefschetz theorem for overconvergent F-iso crystals.
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作者：Yu, Hongjie
摘要：Let X1 be a projective, smooth and geometrically connected curve over Fq with q = p' elements where p is a prime number, and let X be its base change to an algebraic closure of Fq. We give a formula for the number of irreducible B-adic local systems (B =6 p) with a fixed rank over X fixed by the Frobenius endomorphism. We prove that this number behaves like a Lefschetz fixed point formula for a variety over Fq, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne...
