Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

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The proposal is composed of three parts. The first part is aimed at problems in Poisson Lie groups and cluster algebra aspects of representations of quantized universal Lie algebras. The results of this investigation will be used to construct and study invariants of knots and invariants of 3-manifolds. The goal of the second part is the development of the semiclassical quantization of classical field theories. It has two major directions: the first one is the definition of the partition function for space time manifolds with boundaries, the second one is the verification of locality of quantum field theory through the gluing operation for partition functions. This portion of the project is focused mainly on topological gauge theories, in particular on the Chern-Simons theory. The third part of the proposal is focused on problems in equilibrium statistical mechanics. The PI plans to continue to investigate the behavior of correlation functions and of the partition functions for models with fixed boundary conditions on states and, in particular, scaling limits of semiclassical type. Such semiclassical limits appear in the study of correlation functions of `long operators" in the supersymmetric Yang-Mills theory.Significant portion of the first part of the proposal is aimed at providing algebraic tools for constructing quantum topological gauge theories. One of the fundamental problems in particle physics is to construct a theory which explains the dynamics of experimentally detected particles and would explain their variety. The Standard Model is a candidate for such theory. One of the key aspects of this theory is that its classical counterpart has infinite dimensional internal symmetry known as gauge symmetry. This makes the construction of the quantum theory incredibly complicated. Strictly speaking, mathematically the theory is still in its childhood. The second part of the proposal is aimed at resolving such problems in a simpler case of topological gauge theory, where the dynamics is much simpler but the symmetry is exactly the same. The last part of the proposal focuses on the study of phenomena very similar to large deviations in probability theory (estimate the probability that a coin will fall xN times on one side and (1-x)N times on the other side in N trials for large N) and to the stochastic origins of hydrodynamics (we know that the motion of water is deterministic at large scale, but is random at the molecular scale). The PI will continue to study similar phenomena in a number of two dimensional models.