SM: Arithmetic Groups and Their Applications in Combinatorics, Geometry and Topology

4月15日至18日，在夏洛茨维尔的弗吉尼亚大学校园内将举行一个题为“算术群及其在组合学、几何学和拓扑学中的应用”的讲习班。研讨会的科学计划将由来自美国，巴西，法国和以色列的该领域领先专家的讲座组成，并将突出过去几年取得的最重要成果。将特别关注诸如算术群的结构性质的分析等主题（第一Betti数的虚正性，同余子群问题，有界生成），它们在代数几何中的应用（假投影平面和其他重要代数簇的假版本），在微分几何和拓扑学中算术群是一种特殊的群，它的元素是具有整数项的矩阵。这个概念，可以追溯到高斯关于积分二次型的工作，在数学的许多领域中起着至关重要的作用，包括代数和数论的各个部分（例如，自守形式理论（theory of automorphic forms）近年来，算术群理论在代数和微分几何、李理论和组合学中出现了新的应用。讲习班将展示涉及算术组的重要成果。重点将放在使算术组的理论方法，研究人员，特别是年轻的，在各个领域的工作。

A workshop ``Arithmetic Groups and Their Applications in Combinatorics, Geometry and Topology" will be held April 15-18 on the campus of the University of Virginia in Charlottesville. The scientific program of the workshop will be composed of lectures delivered by the leading experts in the area from the US, Brazil, France and Israel, and will highlight the most significant results obtained in the last few years. Special attention will be given to such topics as the analysis of structural properties of arithmetic groups (virtual positivity of the first Betti number, congruence subgroup problem, bounded generation), their applications in algebraic geometry (fake projective planes and fake versions of other important algebraic varieties), in differential geometry and topology (hyperbolic 3-manifolds, isospectral locally symmetricspaces) and in combinatorics (construction of expanders).Arithmetic groups are special groups whose elements are matrices with integral entries. This notion, which can be traced back to the work of Gauss on integral quadratic form, plays a crucial role in many areas of mathematics including algebra and various parts of number theory (e.g., the theory of automorphic forms). In recent years, new applications of the theory of arithmetic groups have emerged in algebraic and differential geometry, Lie theory and combinatorics. The workshop will showcase important results involving arithmetic groups. The focus will be placed on making the methods of the theory of arithmetic groups accessible to researchers, particularly young ones, working in a variety of areas.