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作者：Lawrence, Brian; Venkatesh, Akshay
作者单位：University of Chicago; Institute for Advanced Study - USA
摘要：We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings's original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of p-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems...
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作者：Benoist, Olivier; Wittenberg, Olivier
作者单位：Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS)
摘要：We formulate the real integral Hodge conjecture, a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi-Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel-Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of co...
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作者：Chaika, Jon; Masur, Howard
作者单位：Utah System of Higher Education; University of Utah; University of Chicago
摘要：We show that the set of not uniquely ergodic d-IETs with permutation in the Rauzy class of the hyperelliptic permutation has Hausdorff dimension d - 3/2 [in the (d - 1)-dimension space of d-IETs] for d >= 5. For d = 4 this was shown by Athreya-Chaika and for d is an element of {2, 3} the set is known to have dimension d - 2. This provides lower bounds on the Hausdorff dimension of non-weakly mixing IETs and, with input from Al-Saqban et al. (Exceptional directions for the Teichmuller geodesic ...
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作者：Hesselholt, Lars; Larsen, Michael; Lindenstrauss, Ayelet
作者单位：Nagoya University; University of Copenhagen; Indiana University System; Indiana University Bloomington
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作者：Jelisiejew, Joachim
作者单位：Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences; University of Warsaw
摘要：We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil's Murphy's Law holds up to retraction for this scheme. Our main tool is a generalized version of the Bialynicki-Birula decomposition.
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作者：Erschler, Anna; Zheng, Tianyi
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); University of California System; University of California San Diego
摘要：On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group G we show that its growth vG,S(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-...
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作者：Chen, Dawei; Moeller, Martin; Sauvaget, Adrien; Zagier, Don
作者单位：Boston College; Goethe University Frankfurt; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); CY Cergy Paris Universite
摘要：We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained using intersection theory on strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show that the same results hold for the spin and hyperelliptic components of the strata.
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作者：Bellamy, Gwyn; Craw, Alastair
作者单位：University of Glasgow; University of Bath
摘要：For a finite subgroup Gamma subset of SL(2,C)Gamma \subset \mathrm {SL}(2,\mathbb {C})$$\end{document} and for n >= 1, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity C2/Gamma. It is well known that X:=Hilb[n](S)is a projective, crepant resolution of the symplectic singularity C2n/Gamma n\ where Gamma n=Gamma wreath product Sn is the wreath product. We prove...
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作者：Ding, Jian; Smart, Charles K.
作者单位：University of Pennsylvania; University of Chicago
摘要：We consider a Hamiltonian given by the Laplacian plus a Bernoulli potential on the two dimensional lattice. We prove that, for energies sufficiently close to the edge of the spectrum, the resolvent on a large square is likely to decay exponentially. This implies almost sure Anderson localization for energies sufficiently close to the edge of the spectrum. Our proof follows the program of Bourgain-Kenig, using a new unique continuation result inspired by a Liouville theorem of Buhovsky-Logunov-...
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作者：Behrens, Mark; Rezk, Charles
作者单位：University of Notre Dame; University of Illinois System; University of Illinois Urbana-Champaign
摘要：We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable v(n)-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computation...
