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作者：Killip, Rowan; Visan, Monica
摘要：We prove global well-posedness of the Korteweg-de Vries equation for initial data in the space H-1 (R). This is sharp in the class of H-s (R) spaces. Even local well-posedness was previously unknown for s < -3/4. The proof is based on the introduction of a new method of general applicability for the study of low-regularity well-posedness for integrable PDE, informed by the existence of commuting flows. In particular, as we will show, completely parallel arguments give a new proof of global wel...
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作者：Moreira, Joel; Richter, Florian K.; Robertson, Donald
摘要：In this paper we show that every set A subset of N with positive density contains B +C for some pair B, C of infinite subsets of N, settling a conjecture of Erdos. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.
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作者：Shmerkin, Pablo
摘要：We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the L-q dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of xp- and xq-invariant se...
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作者：Khayutin, Ilya
摘要：We prove the mixing conjecture of Michel and Venkatesh for toral packets with negative fundamental discriminants and split at two fixed primes, assuming all splitting fields have no exceptional Landau-Siegel zero. As a consequence we establish for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all of whose components have the same fundamental discriminant, assuming the CM fields are split at two fixed primes and have no e...
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作者：Larsen, Michael; Shalev, Aner; Tiep, Pham Huu
摘要：The probabilistic Waring problem for finite simple groups asks whether every word of the form w(1)w(2) , where w(1) and w(2) are non-trivial words in disjoint sets of variables, induces almost uniform distributions on finite simple groups with respect to the L-1 norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distributions on finite simple groups of Lie type of bounded rank, and study relate...
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作者：Brendle, Simon
摘要：We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates that ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension n >= 12, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension 3. In the second part, we develop a theory of ancient solu...
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作者：Binyamini, Gal; Novikov, Dmitry
摘要：We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For delta is an element of (0, 1) and a complex cell C, we define its holomorphic extension C subset of C-delta, which is again a complex cell. The hyperbolic geometry of C within C-delta provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry....
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作者：Bordenave, Charles; Collins, Benoit
摘要：Let (sigma(1), . . . , sigma(d)) be a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as n -> infinity, these permutations viewed as operators on the n-1 dimensional vector space {x(1), . . . , x(n)) is an element of C-n, Sigma x(i) = 0}, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refine...
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作者：Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin
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作者：Mok, Ngaiming; Pila, Jonathan; Tsimerman, Jacob
摘要：We prove the Ax-Schanuel theorem for a general (pure) Shimura variety. A basic version of the theorem concerns the transcendence of the uniformization map from a bounded Hermitian symmetric space to a Shimura variety. We then prove a version of the theorem with derivatives in the setting of jet spaces, and finally a version in the setting of differential fields. Our method of proof builds on previous work, combined with a new approach that uses higher-order contact conditions to place varietie...
