From Representation Theory to Mathematical Physics and Back

Cuny Medgar Evers College, Brooklyn NY

Investigators

Abstract

The theme of the conference "From Representation to Mathematical Physics and Back" is the continued influence of mathematics and physics on each other. This symbiotic relationship has played a significant role in these disciplines for centuries. For instance, Newton's discovery of gravity necessitated his development of calculus. While the interactions between mathematics and physics have evolved over time, some of the deepest aspects of these fields are still intertwined. In particular, the discovery of quantum physics in the twentieth century led mathematicians to study difficult problems in analysis, topology, and algebra. In turn, solutions to these questions have then been reinterpreted by physicists, and have led to new physics questions. Understanding how gravity fits into our quantum mechanical universe remains one of the most important open questions in physics and mathematics. Recent advances in these areas will be presented at this conference. A primary goal of the meeting is to introduce a young generation of researchers, including those from typically underrepresented groups, to these fundamental mathematical and physical problems. For more information see the conference website: http://scgp.stonybrook.edu/archives/34420 The conference will highlight progress in the areas of vertex operator algebras, conformal field theory, categorification, low dimensional topology and representation theory of affine Lie algebras, loop groups, and quantum groups. String theory gave rise to the mathematical theory of vertex operator algebras, which led to the construction of representations of affine Lie algebras and the Moonshine module of the Monster group. These mathematical constructions have in turn led to ideas about 3-dimensional quantum gravity. In another direction, the discovery of the Jones polynomial led to a physical construction of 3-dimensional TQFTs, which in turn led to many mathematical discoveries in quantum groups and low dimensional topology. Crane and Frenkel introduced the categorification program with the goal of upgrading 3-dimensional TQFTs coming from representation theory of quantum groups to 4-dimensional TQFTs. They suggested that one should try to construct categorical quantum group actions to achieve this aim. These ideas led to the discovery of Khovanov homology. Other link homologies have been constructed from representation-theoretic, algebraic-geometric, combinatorial, and physical structures. Consequently, categorification has become a truly interdisciplinary subject. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.