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• Nonabelian level structures, Nielsen equivalence, and Markoff triples AI翻译
  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...
• On L∞ estimates for fully non-linear partial differential equations AI翻译
  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...
• Uniform character bounds for finite classical groups AI翻译
  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...
• Algebraic integers with conjugates in a prescribed distribution AI翻译
  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...
• A reverse Minkowski theorem AI翻译
  作者：Regev, Oded; Stephens-Davidowitz, Noah
  摘要：We prove a conjecture due to Dadush, showing that if L subset of R-n is a lattice such that det (L ') >= 1 for all sublattices L 'subset of L, then & sum;(e-pi t2 parallel to y parallel to 2 <= 3/2, )(y is an element of L)where t:=10(logn+2). From this we derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski's celebrated first theorem. We also derive a bound on the covering radius.
• Log-concave polynomials II: High-dimensional walks and an FPRAS for counting bases of a matroid AI翻译
  作者：Anari, Nima; Liu, Kuikui; Gharan, Shayan Oveis; Vinzant, Cynthia
  摘要：We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0 < q < 1. Consequently, we can sample random spanning forests in a graph and estimate the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has edge expansion at least 1. Our algorithm and proof...
• Purity for flat cohomology AI翻译
  作者：Cesnavicius, Kestutis; Scholze, Peter
  摘要：We establish the flat cohomology version of the Gabber-Thomason purity for & eacute;tale cohomology: for a complete intersection Noetherian local ring (R,m) and a commutative, finite, flat R-group G, the flat cohomology H-m(i)(R,G) vanishes for for i <= dim(R). For small i, this settles conjectures of Gabber that extend the Grothendieck-Lefschetz theorem and give purity for the Brauer group for schemes with complete intersection singularities. For the proof, we reduce to a flat purity statemen...
• A counterexample to the periodic tiling conjecture AI翻译
  作者：Greenfeld, Rachel; Tao, Terence
  摘要：The periodic tiling conjecture asserts that any finite subset of a lattice Zd that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large d, which also implies a disproof of the corresponding conjecture for Euclidean spaces Rd. In fact, we also obtain a counterexample in a group of the form Z2 x G0 for some finite abelian 2-group G0. Our methods rely on encoding a Sudoku puzzle whose rows and other non-horizontal lines ar...
• Symplectic monodromy at radius zero and equimultiplicity of μ-constant families AI翻译
  作者：De Bobadilla, Javier Fernandez; Pelka, Tomasz
  摘要：We show that every family of isolated hypersurface singularities with constant Milnor number has constant multiplicity. To achieve this, we endow the A'Campo model of radius zero monodromy with a symplectic structure. This new approach allows us to generalize a spectral sequence of McLean converging to fixed point Floer homology of iterates of the monodromy to a more general setting that is well suited to study mu -constant families.
• 1Corrigendum: Marginal triviality of the scaling limits of critical 4D Ising and φ44 models(vol 194,pg 163, 2021) AI翻译
  作者：Aizenman, Michael; Duminil-Copin, Hugo