Groups and Geometries (12w5034)

The Banff International Research Station will host the "Groups and Geometries" workshop from September 2nd to September 7th, 2012.

Group theory is an abstract subject, but it is also the mathematical way to investigate symmetry. Thus many problems fom various areas of mathematics which involve some sort of symmetry are susceptible to the application of group theory. In this proposal we plan to cover three main areas: the abstract structure of finite groups; some geometrical aspects of group theory; and several recent applications of group theory to other areas of mathematics.



The basic building blocks of all finite groups are the simple groups, and the finite simple groups were classified in one of the most monumental mathematical achievements of the 20th century, involving many researchers over a period of many years. A major effort is currently under way to provide a new, self-contained proof of the classification, and a number of the proposed participants are involved. Reports on recent work, and plans for attacks on the remaining parts of this vital program, will be one of the major themes of the meeting. Another theme will be the connections between groups and geometrical structures such as Bruhat-Tits buildings; here the groups involved are not just finite simple groups, but many other types of algebraic and arithmetic groups. The third theme will be some spectacular recent applications of group theory. One such concerns the topic of expansion in groups, and its applications range from the construction of expander graphs to the development of new sieve methods in analytic number theory.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).