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作者：Hill, Michael; Lawson, Tyler
作者单位：University of Virginia; University of Minnesota System; University of Minnesota Twin Cities
摘要：The cohomology theory known as , for topological modular forms, is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from with level structure to forms of -theory. In parti...
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作者：de Fernex, Tommaso; Docampo, Roi
作者单位：Utah System of Higher Education; University of Utah; Universidade Federal Fluminense
摘要：Let be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over . We prove that every terminal valuation over is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of . In dimension two, this result gives a new proof of the theorem of Fernandez de Bobadilla and Pe Pereira stating that, for surfaces...
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作者：Shin, Sug Woo; Templier, Nicolas
作者单位：University of California System; University of California Berkeley; Korea Institute for Advanced Study (KIAS); Cornell University
摘要：We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let be a reductive group over a number field which admits discrete series representations at infinity. Let be the associated -group and a continuous homomorphism which is irreducible and does not factor through . The families under consideration consist of discrete automorphic representations of of given weight and level and we let either the weight or the le...
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作者：Carlotto, Alessandro; Schoen, Richard
作者单位：Swiss Federal Institutes of Technology Domain; ETH Zurich; University of California System; University of California Irvine
摘要：We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of N-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large T we can engineer solutions where any two massive bodies do not interact at all for any time , in striking contras...
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作者：Hakavuori, Eero; Le Donne, Enrico
作者单位：University of Jyvaskyla
摘要：We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subriemannian geodesics.
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作者：Gong, Xianghong; Stolovitch, Laurent
作者单位：University of Wisconsin System; University of Wisconsin Madison; Universite Cote d'Azur; Centre National de la Recherche Scientifique (CNRS); Universite Cote d'Azur
摘要：We study a germ of real analytic n-dimensional submanifold of that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions, we show its equivalence to a normal form under a local biholomorphism at the singularity. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we investigate the existence of a complex submanifold of positive di...
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作者：Cha, Jae Choon; Powell, Mark
作者单位：Pohang University of Science & Technology (POSTECH); Korea Institute for Advanced Study (KIAS); University of Quebec; University of Quebec Montreal
摘要：We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps, data which is sufficient for a topological disc to exist...
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作者：Brendle, Simon; Huisken, Gerhard
作者单位：Stanford University; Eberhard Karls University of Tubingen
摘要：We define a notion of mean curvature flow with surgery for two-dimensional surfaces in with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author (Invent Math, 2015).
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作者：Bernstein, Jacob; Wang, Lu
作者单位：Johns Hopkins University; University of Wisconsin System; University of Wisconsin Madison
摘要：The entropy is a natural geometric quantity which measures the complexity of a hypersurface in . It is non-increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow. In (Colding et al., J Differ Geom 95(1):53-69, 2013), Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinking solutions of the mean curvature flow in , the entropy is uniquely minimized at the round sphere. They conjectured that, for , the r...
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作者：Vasseur, Alexis F.; Yu, Cheng
作者单位：University of Texas System; University of Texas Austin
摘要：In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins (Commun Math Phys 238:211-223 2003) entropy conservation. The main contribution of this paper is to derive the Mellet and Vasseur (Commun Partial Differ Equ 32:431-452, 2007) type inequality for weak solutions, even if it is not verified by the first level of approximation. This provides existence of global soluti...
