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作者：Guth, Larry; Wang, Hong; Zhang, Ruixiang
摘要：We prove a sharp square function estimate for the cone in R-3 and consequently the local smoothing conjecture for the wave equation in 2 + 1 dimensions.
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作者：Koukoulopoulos, Dimitris; Maynard, James
摘要：Let psi : N -> R->= 0 be an arbitrary function from the positive integers to the non-negative reals. Consider the set A of real numbers a for which there are infinitely many reduced fractions a/q such that vertical bar alpha-a/q vertical bar <= psi(q)/q. If Sigma(infinity)(q=1) psi(q)phi(q)/q = infinity, we show that A has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the i...
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作者：DeMarco, Laura; Krieger, Holly; Ye, Hexi
摘要：We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on P-1((Q) over bar). We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over C. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Ma...
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作者：Goette, S.; Kerin, M.; Shankar, K.
摘要：In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.
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作者：Rizzardo, Alice; Van den Bergh, Michel
摘要：In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field.
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作者：Brunat, Olivier; Dudas, Olivier; Taylor, Jay
摘要：We show that the decomposition matrix of unipotent l-blocks of a finite reductive group G(F-q) has a unitriangular shape, assuming q is a power of a good prime and l is very good for G. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand-Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity ...
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作者：Balister, Paul; Bollobas, Bela; Morris, Robert; Sahasrabudhe, Julian; Tiba, Marius
摘要：We show that there exist absolute constants Delta > delta > 0 such that, for all n >= 2, there exists a polynomial P of degree n, with coefficients in { -1, 1}, such that delta root n <= vertical bar P(z)vertical bar <= Delta root n for all z is an element of C with vertical bar z vertical bar = 1. This confirms a conjecture of Littlewood from 1966.
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作者：Schafhauser, Christopher
摘要：It is shown that if A is a separable, exact C*-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF algebra with a unique trace. Modulo the UCT, this provides an abstract characterization of C*-subalgebras of simple, unital AF-algebras. As a consequence, for a countable, discrete, amenable group G acting on a second countable, locally compact, Hausdorff space X, C-o(X) infinity(r)...
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作者：Chodosh, Otis; Mantoulidis, Christos
摘要：The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show that for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with m...
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作者：Browning, Tim; Sawin, Will
摘要：We develop a geometric version of the circle method and use it to compute the compactly supported cohomology of the space of rational curves through a point on a smooth affine hypersurface of sufficiently low degree.
