Conference Proposal: Unity in Mathematics

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Principal Investigator: Pavel Etingof, Joseph Harris, Isadore SingerProposal Number: DMS-0315184Institution: Massachusetts Institute of Technology Title: Unity in MathematicsAbstract:The goals of the conference "Unity of mathematics" are to discuss important recent developments in mathematics, to chart promising new directions, and to focus on connections between different fields. Specifically, the conference will emphasize two areas where there have been significant advances in recent years, namely 1) geometry and physics, and 2) representation theory. The discussion of advances in geometry and physics will focus on new ideas coming from quantum field theory and string theory, and their geometric consequences. This discussion will involve diverse topics, such as mirror symmetry, Seiberg-Witten theory, twisted K-theory, chiral algebras, noncommutative Yang-Mills theory, formality theory, intersection theory on moduli spaces, etc. The discussion of advances in representation theory will focus on the interactions between representation theory and geometry; these interactions have recently led to the development of powerful new methods of studying representations. Some of the methods are the geometric Langlands program, quiver varieties, geometric realizations of quantum groups and their representations. The conference ``Unity of Mathematics'' is designed to bring together a diverse group of leading senior mathematicians and young researchers for a discussion of the most important recent developments in mathematics, with the focus on the topics "geometry and physics", and "representation theory". The first topic involves developments in geometry that were triggered by recent breakthroughs in quantum field theory (which describes the motion of elementary particles) and string theory (which attempts to unify quantum field theory and Einstein's general relativity). Although string theory has not yet made successful experimental predictions, it is striking that it did make successful predictions in geometry. These predictions are the first example in the history of science when geometry served as a ``lab'' for theoretical physics, and provide as much evidence that string theory is ``correct'' as an experimental verification would be. The second topic involves developments in representation theory (a branch of algebra that could be very roughly defined as spectroscopy of symmetry), which have resulted from systematic use of deep ideas from geometry and topology (theory of shape). The conference will attract numerous mathematicians working in different fields, as well as many theoretical physicists. Young researchers are expected to find the conference especially useful because it will discuss developments occurring at boundaries between fields. This may well be an epiphany to those trained in a narrow specialty. The meeting will be a unique occasion; prominent mathematicians will not only review recent advances but will discuss their broad implications for mathematics. Special provisions will be made to encourage young researchers, graduate students, women and minorities to participate.