The main goal of the proposed research is to study the physical realization of various constructions in geometric representation theory, notably the physical framework for knot homologies and the geometric Langlands program. The physical realization is based on topological gauge theory and topological string theory which describe the supersymmetric sector of the physical gauge theory/string theory. Recently the idea of "categorification" led to a number of remarkable developments in various branches of mathematics which may well have a physical interpretation. These include a variety of problems in representation theory, as well as "categorification" of polynomial knot invariants in low-dimensional topology. Related structures recently emerged in physics. In four-dimensional gauge theory the study has led to new interesting connections with the geometric Langlands program. The realization of Chern-Simons invariants of knots and links in terms of BPS states in the dual string theory led to formulation of triply-graded knot homologies. These developments point to many new connections between various problems in physics and mathematics that are described in this proposal. Broader impact: This proposal, on the one hand, should advance the understanding of gauge theory dynamics and string theory and, on the other hand, lead to important lessons and new results in related areas of mathematics, including enumerative geometry, homological algebra, low-dimensional topology, and representation theory. The PI is actively involved in organizing seminars, workshops and conferences on various topics related to the proposed research, and in mentoring students at all levels.
