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作者：Du, Xiumin; Guth, Larry; Li, Xiaochun
摘要：We show that lim(t -> 0) e(it Delta) f(x) = f(x) almost every where for all f is an element of H-s (R-2) provided that s > 1/3. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.
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作者：Marks, Andrew S.; Unger, Spencer T.
摘要：We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If k >= 1 and A, B subset of R-k are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than k, then A and B are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a rec...
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作者：Walker, Mark E.
摘要：The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that the ith Betti number beta(i)(M) of a nonzero module M of finite length and finite projective dimension over a local ring R of dimension d should be at least ((d)(i)). It would follow from the validity of this conjecture that Sigma(i)beta(i)(M)>= 2d. We prove the latter inequality holds in a large number of cases and that, when R is a complete intersection in which 2 is invertible, equality holds if and only if M is isomorphic to the quot...
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作者：Fasel, Jean
摘要：We explain a mistake that occurred in the proof of Murthys conjecture by the author.
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作者：Wang, Guozhen; Xu, Zhouli
摘要：We prove that the 2-primary pi(61) is zero. As a consequence, the Kervaire invariant element theta(5) is contained in the strictly defined 4-fold Toda bracket < 2, theta(4), theta(4) 2 >. Our result has a geometric corollary: the 61-sphere has a unique smooth structure, and it is the last odd dimensional case - the only ones are S-1, S-3, S-5 and S-61. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential d(3)(D-3) = B-3. We prov...
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作者：Bonnafe, Cedric; Dat, Jean-Francois; Rouquier, Raphael
摘要：This paper is a continuation and a completion of the work of the first and the third author on the Jordan decomposition. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in nondescribing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group this is the modular version of a fundamental result of Lusztig, and the best approximation of the character -theoretic Jordan decomposition that can be obt...
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作者：Grabowski, Lukasz; Mathe, Andras; Pikhurko, Oleg
摘要：Laczkovich proved that if bounded subsets A and B of R-k have the same nonzero Lebesgue measure and the upper box dimension of the boundary of each set is less than k, then there is a partition of A into finitely many parts that can be translated to form a partition of B. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaring and Hilbert's third pro...
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作者：Dospinescu, Gabriel; Le Bras, Arthur-Cesar
摘要：We describe the de Rham complex of the etale coverings of Drinfeld's p-adic upper half-plane for GL(2)(Q(p)). Conjectured by Breuil and Strauch, this description gives a geometric realization of the p-adic local Langlands correspondence for certain two-dimensional de Rham representations of Gal((Q(p)) over bar /Q(p)).
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作者：Zhu, Xinwen
摘要：We endow the set of lattices in Q(P)(n) with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake equivalence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.
