Global/Local Conjectures in Representation Theory of Finite Groups (11w5008)
Gunter Malle (Technische Universität Kaiserslautern)
Gabriel Navarro (Universidad de Valencia)
Pham Tiep (University of Arizona)
The Representation Theory of Finite Groups is a thriving subject, with many fascinating and deep open problems, and significant recent successes. In 1963 Richard Brauer formulated a list of deep conjectures about ordinary and modular representations of finite groups. These have led to many new concepts and methods, but basically all of his main conjectures are still unsolved to the present day. A new development was opened up by John McKay in 1972 with an observation on degrees of characters for simple groups which was soon named the McKay conjecture and led to a wealth of difficult problems, relating global and local properties of finite groups, that also remain unproved. Profoundly significant are the Alperin Weight Conjecture formulated by J. Alperin in 1986, and the structural explanation of some of the conjectures proposed by M. Broue'. We all gather that there should be a hidden theory explaining all these phenomena, but we are unable to find it yet. Recently, considerable progress has been made by the proposers and others in several directions. Firstly, there is now a reduction to simple groups of the original McKay conjecture. Still, more and deeper knowledge of simple groups is needed in order to give a proof of these conjectures. Another line of successes has been provided by proving some of these statements in some special cases. The difficulty of those give us a hint of the task ahead. For instance, Brauer's k(GV)-problem has only been solved after years of inspiring work of many mathematicians, including the Fields medalist John Thompson.
The aim of the proposed meeting is to bring together the leading experts on modular representation theory of finite groups to exploit the substantial recent advances on some of these fundamental conjectures, including the McKay conjecture and its refinements, the Alperin Weight Conjecture, the Brauer Height Zero Conjecture, and the Broue' conjecture. This will also be a unique opportunity for young mathematicians to learn about these exciting developments and become directly involved in this fascinating area of research. It is our hope that the meeting will strengthen existing and foster new collaborations, and facilitate significant progress along the lines of all these important conjectures.