Mathematical Sciences: Topology, Geometry and Physics

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9626698 Freed This grant supports the global analysis and differential geometry group at the University of Texas at Austin. The proposed research concentrates on the analytic and algebraic aspects of mathematical physics. Research topics to be pursued include: Chern-Simons theory and the structure of topological quantum field theory; infinite dimensional integrable systems and nonlinear Schrodinger equations; symbolic dynamics and branched manifolds; geometrical structures in nonrelativisitic quantum mechanics. The project also includes educational activities involving undergraduate students and high school students. Quantum field theory is an active area within theoretical and mathematical physics: ultimately, the researchers in this field hope to come up with a rigorous and unifying framework explaining various basic forces of nature. Integrable systems constitute an important subarea within the theory of dynamical systems - a dynamical system is given by a system of ordinary differential equations; in such a system the rate at which the system evolves over time is independent of time. Dynamical systems are used to model various natural phenomena such as population growth and weather systems.