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作者：Koenigsmann, Jochen
摘要：We show that Z is definable in Q by a universal first-order formula in the language of rings. We also present an for all there exists-formula for Z in Q with just one universal quantifier. We exhibit new diophantine subsets of Q like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of nonsquares is diophantine.
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作者：Brendle, Simon
摘要：We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution of mean curvature flow in R-3 with genus 0. More generally, we show that the only properly embedded self-similar shrinkers in R-3 with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.
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作者：Duke, W.; Imamoglu, O.; Toth, A.
摘要：To an ideal class of a real quadratic field eve associate a certain surface. This surface, which is a new geometric invariant, has the usual modular closed geodesic as its boundary. Furthermore, its area, is determined by the length of an associated backward continued fraction. We study the distribution properties of this surface on average over a genus. In the process we give an extension and refinement of the Katok-Sarnak formula.
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作者：Reiher, Christian
摘要：Turan's theorem is a cornerstone of extremal graph theory. It asserts that for any integer r >= 2, every graph on n vertices with more than r-2/2(r-1) . n(2) edges contains a clique of size r, i.e., r mutually adjacent vertices. The corresponding extremal graphs are balanced (r - 1)-partite graphs. The question as to how many such r-cliques appear at least in any n-vertex graph with gamma n(2) edges has been intensively studied in the literature. In particular, Lovasz and Simonovits conjecture...
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作者：Schmidt, Alexander; Stix, Jakob
摘要：Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with (tale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field k that is finitely generated over Q has a fundamental system of (affine) anabelian Zariski-neighborhoods. This was predicted by Grothendieck in his letter to Faltings.
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作者：De Lellis, Camillo; Spadaro, Emanuele
摘要：This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multiple valued functions.
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作者：Treumann, David; Venkatesh, Akshay
摘要：If sigma is an automorphism of order p of the semisimple group G, there is a natural correspondence between mod p cohomological automorphic forms on G and G(sigma). We describe this correspondence in the global and local settings.
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作者：De Lellis, Camillo; Spadaro, Emanuele
摘要：This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely, the construction of a center manifold, i.e., an approximate average of the sheets of an almost flat area minimizing current. Such a center manifold is accompanied by a Lipschitz multivalued map on its normal bundle, which approximates the current with a high degree of accuracy. In the third an...
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作者：Schiffmann, Olivier
摘要：We prove that the number of geometrically indecomposable vector bundles of fixed rank r and degree d over a smooth projective curve X defined over a finite field is given by a polynomial (depending only on r; d and the genus g of X) in the Weil numbers of X. We provide a closed formula - expressed in terms of generating series- for this polynomial. We also show that the same polynomial computes the number of points of the moduli space of stable Higgs bundles of rank r and degree d over X. This...
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作者：Kiselev, Alexander; Ryzhik, Lenya; Yao, Yao; Zlatos, Andrej
摘要：It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law that is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars that interpolates between these two equations, via a parameter alpha epsilon [0, 1/2] appearing in th...
