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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：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：Moeller, Sven; Scheithauer, Nils R.
摘要：We prove a dimension formula for the weight-1 subspace of a vertex operator algebra Vorb(g) obtained by orbifolding a strongly rational, holo-morphic vertex operator algebra V of central charge 24 with a finite-order automorphism g. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in Aut(V ).Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra V Lambda ...
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作者：Schroeer, Stefan
摘要：We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite e ' tale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine and Abrashkin for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on ...
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作者：Baldi, Gregorio; Ullmo, Emmanuel
摘要：Let Gamma subset of PU(1, n) be a lattice and Sr be the associated ball quotient. We prove that, if Sr contains infinitely many maximal complex totally geodesic subvarieties, then Gamma is arithmetic. We also prove an Ax-Schanuel Conjecture for Sr, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise Sr inside a period domain for polarised integral variations of Hodge structure and interpret totally geodesic subvarieties as unli...
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作者：Khot, Subhash; Minzer, Dor; Safra, Muli
摘要：We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the 2-to-2 Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. 162 (2005), 439-485], and new hardness gaps for Unique-Games. The Grassmann graph Grglobal contains induced subgraphs Gr(local) that are themselves isomorphic to Grassmann gra...
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作者：Fraczyk, Mikolaj; Gelander, Tsachik
摘要：We prove the following conjecture of Margulis. Let G be a higher rank simple Lie group, and let ? <= G be a discrete subgroup of infinite covol-ume. Then, the locally symmetric space ?\G/K admits injected balls of any radius. This can be considered as a geometric interpretation of the cel-ebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We ...
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作者：Wendl, Chris
摘要：We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomor-phic curves are generically regular, and simple index 0 curves in dimen-sions greater than four are generically super-rigid, implying, e.g., that the Gromov-Witten invariants of Calabi-Yau 3-folds reduce to sums of local in-variants for finite sets of embedded curves. We also establish ...
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作者：Dasgupta, Samit; Kakde, Mahesh
摘要：Let H/F be a finite abelian extension of number fields with F totally real and H a CM field. Let S and T be disjoint finite sets of places of F satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element Theta H/F S,T annihilates the T-smoothed class group ClT (H). We prove this conjecture away from p = 2, that is, after tensoring with Z[1/2]. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting id...
