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作者：Gowers, W. T.; Green, Ben; Manners, Freddie; Tao, Terence
摘要：We prove a conjecture of K. Marton, widely known as the polynomial Freiman-Ruzsa conjecture, in characteristic 2. The argument extends to odd characteristic, with details to follow in a subsequent paper.
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作者：Heikkila, Susanna; Pankka, Pekka
摘要：We show that, if a closed, connected, and oriented Riemannian n-manifold N admits a non-constant quasiregular mapping from the Euclidean nspace Rn, then the de Rham cohomology algebra H & lowast;dR(N) of N embeds into the exterior algebra n & lowast;Rn. As a consequence, we obtain a homeomorphic classification of closed simply connected quasiregularly elliptic 4-manifolds.
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作者：Bhargava, Manjul
摘要：Of the (2H+1)(n) monic integer polynomials f(x)=x(n)+a1x(n-1)+...+a(n) with max{|a(1)|,...,|a(n)|} <= H, how many have associated Galois group that is not the full symmetric group S-n?There are clearly >> Hn-1 such polynomials, as may be obtained by setting a(n)=0. In 1936, van der Waerden conjectured that O(Hn-1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n <= 4, due to work of van der Waerden and Chow...
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作者：Shankar, Ravi; Yuan, Yu
摘要：We derive a priori interior Hessian estimates and interior regularity for the sigma 2 equation in dimension four. Our method provides both a new proof for the corresponding three dimensional results and a Hessian estimate for smooth solutions satisfying a dynamic semi-convexity condition in higher (n >= 5) dimensions.
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作者：Ruhstorfer, Lucas
摘要：In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture and Brauer's height zero conjecture for the prime 2.
