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作者：Freitas, Nuno; Le Hung, Bao V.; Siksek, Samir
作者单位：University of Bayreuth; Harvard University; University of Warwick
摘要：We prove that all elliptic curves defined over real quadratic fields are modular.
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作者：Sheridan, Nick
作者单位：Princeton University; Institute for Advanced Study - USA
摘要：We prove Homological Mirror Symmetry for a smooth -dimensional Calabi-Yau hypersurface in projective space, for any (for example, is the quintic threefold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the '-dimensional pair of pants'; the introduction of the 'relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an 'orbifold' Fukaya ca...
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作者：Voisin, Claire
作者单位：Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique; Institut Polytechnique de Paris; Ecole Polytechnique
摘要：We prove that the general quartic double solid with nodes does not admit a Chow theoretic decomposition of the diagonal, (or equivalently has a nontrivial universal group,) and the same holds if we replace in this statement Chow theoretic by cohomological. In particular, it is not stably rational. We also deduce that the general quartic double solid with seven nodes does not admit a universal codimension cycle parameterized by its intermediate Jacobian, and even does not admit a parametrizatio...
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作者：Miclo, Laurent
作者单位：Universite Federale Toulouse Midi-Pyrenees (ComUE); Universite de Toulouse; Institut National des Sciences Appliquees de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier
摘要：Consider an ergodic Markov operator M reversible with respect to a probability measure mu on a general measurable space. It is shown that if M is bounded from L-2(mu) to L-p(mu), where p > 2, then it admits a spectral gap. This result answers positively a conjecture raised by Hoegh-Krohn and Simon (J. Funct. Anal. 9:121-80, 1972) in the more restricted semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order forweighted finit...
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作者：Manners, Freddie
作者单位：University of Oxford
摘要：The pyjama stripe is the subset of consisting of a vertical strip of width around every integer -coordinate. The pyjama problem asks whether finitely many rotations of the pyjama stripe around the origin can cover the plane. The purpose of this paper is to answer this question in the affirmative, for all positive . The problem is reduced to a statement closely related to Furstenberg's Theorem from topological dynamics, and is proved by analogy with that result.
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作者：Kozma, Gady; Nitzan, Shahaf
作者单位：Weizmann Institute of Science; University System of Ohio; Kent State University; Kent State University Salem; Kent State University Kent
摘要：We show that any finite union of intervals supports a Riesz basis of exponentials.
