Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The proposal is focused on problems which lie at the interface of representation theory and mathematical physics. More specially, it aims at problems in the representation theory of quantum groups at roots of 1, on invariants of 3-manifolds, certain problems in local quantum field theory (such as the construction of perturbative Chern-Simons theory), and on problems in integrable systems and solvable models of statistical mechanics.One of the central questions in modern theoretical physics is the construction of the model of fundamental interaction which is consistent with the experiment and mathematically adequate. The framework of local quantum field theory is main concept behind the standard model. However the framework of quantum field theory still largely remain a mathematical puzzle. Part of my research will focus on understanding this puzzle in the context of semi-classical quantization. The goal of other parts of the proposal is the construction of topological and integrable quantum field theories combinatorically and the study of emerging algebraic and analytical problems.