Cohomology, Representations, and Coverings of Curves

点击翻译按钮获取中文摘要

The mathematical concept of a group, a set together with an operation for combining two elements to produce another element, plays a central role in mathematics and its applications because it encodes intuitive notions of symmetry in a precise manner. The classification of the collection of finite groups known as finite simple groups is a monumental achievement of modern mathematics that has led to a revolution in using group theory to study other fields. The utility of group theory has been greatly expanded by advances in computation. One goal of the project is to describe useful presentations of finite simple groups that lead to more computational efficiency. This will lead to solutions of fundamental problems in number theory. Another goal of the project is to study a group-theoretical problem that will lead will lead to results showing the existence and construction of expander graphs, which have been of great importance in computer science.In more detail, one goal of the project is to prove a generalization of the fact that if H is a finitely generated Zariski dense subgroup of a semisimple algebraic group, then H contains a strongly dense subgroup. This will give some new results about superstrong approximation in number theory and results on expander graphs. A second goal of the project is to prove the conjecture that every finite simple group has a presentation with two generations and at most four relations. This will lead to advances in computational number theory. A third goal of the project is to classify generic stabilizers for simple algebraic groups in irreducible linear representations. This will fit into the program of Bhargava to solve classification problems of algebraic families. A fourth goal of the project is to classify monodromy groups of coverings of low genus Riemann surfaces.