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作者：Haag, Summer; Kertzer, Clyde; Rickards, James; Stange, Katherine E.
摘要：In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes appears as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski clo...
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作者：Kollar, Janos; Voisin, Claire
摘要：We prove in this paper the smoothability of cycles modulo rational equivalence below the middle dimension, that is, when the dimension is strictly smaller than the co dimension. We introduce and study the class of cycles obtained as flat pushforwards of Chern classes (or equivalently, flat pushforwards of products of divisors) and prove that they are smooth- able below the middle dimension. Our main result is that all cycles (of any dimension) on a smooth projective variety are flat pushforwar...
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作者：Chen, William Y.
摘要：In this paper we establish a congruence on the degree of the map from a component of a Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves. Combinatorially, this can be expressed as a congruence on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this congruence to show that for all but finitely many primes p, the group of Markoff automorphisms acts transitively o...
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作者：Guo, Bin; Phong, Duong H.
摘要：Sharp LO degrees estimates are obtained for general classes of fully non-linear PDE's on non-Kahler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the Kahler case. The key idea is still a comparison with an auxiliary Monge-Ampe`re equation, but this time on a ball with Dirichlet boundary conditions, so that it always admits a unique solution. The method applies not just to compact Hermitian manifolds, but also to the Dirichlet problem, to op...
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作者：Poltoratski, A.
摘要：We prove pointwise convergence for the scattering data of a Dirac system of differential equations. Equivalently, we prove an analog of Carleson's theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. Our proofs are based on the study of resonances of Dirac systems using families of meromorphic inner functions, generated by a Riccati equation corresponding to the system.
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作者：Landesman, Aaron; Litt, Daniel
摘要：Let Sigma(g,n) be an orientable surface of genus g with n punctures. We study actions of the mapping class group Mod(g,n) of Sigma(g,n) via Hodge -theoretic and arithmetic techniques. We show that if rho : pi(1)(Sigma(g,n)) -> GL(r)(C) is a representation whose conjugacy class has finite orbit under Mod(g,n), and r < root g + 1, then rho has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman -Wieland conjecture, the Fonta...
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作者：Kwan, Matthew; Sah, Ashwin; Sawhney, Mehtaab; Simkin, Michael
摘要：We prove a 1973 conjecture due to Erd & odblac;s on the existence of Steiner triple systems with arbitrarily high girth.
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作者：Larsen, Michael; Tiep, Pham Huu
摘要：For every finite quasisimple group of Lie type G , every irreducible character chi of G , and every element g of G , we give an exponential upper bound for the character ratio chi ( g ) /chi (1) with exponent linear in log|G| | G | g G , or, equivalently, in the ratio of the support of g to the rank of G . We give several applications, including a proof of Thompson's conjecture for all sufficiently large simple symplectic groups, orthogonal groups in characteristic 2, and some other infinite f...
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作者：Dolgopyat, D.; Kanigowski, A.; Hertz, F. Rodriguez
摘要：Let f be a C 1+ alpha diffeomorphism of a compact manifold M preserving a smooth measure mu . We show that if f : ( M, mu ) -> ( M, mu ) is exponentially mixing, then it is Bernoulli.
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作者：Smith, Alexander
摘要：Given a compact subset Sigma of R obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in Sigma. The distribution of conjugates of such an integer defines a probability measure on Sigma; our main result gives a necessary and sufficient condition for a given probability measure on Sigma to be the limit of some sequence of distributions of conjugates. As one consequence, we show there are infinitely many totally positive algebraic integers alpha w...
