Symplectic geometry, physics and algebraic combinatorics

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AbstractAward: DMS-0093647Principal Investigator: Christopher T. WoodwardThe principal investigator plans to develop the theory of groupactions on symplectic manifolds and geometric invariant theory,and extend applications to certain 2d conformal field theoriesand algebraic combinatorics. In particular, he will investigatethe theory of loop group actions, group-valued moment maps, andrelations to the Wess-Zumino-Witten model in conformal fieldtheory and cohomology pairings on moduli spaces of flatconnections; the asymptotic limit of the quantum 6j symbols; anddecomposition of products of conjugacy classes in disconnectedcompact groups and twisted forms of quantum cohomology. He willintegrate his research with a variety of educational projects,such as dynamical systems in the calculus and linear algebrasequence at Rutgers; strengthening the joint major with computerscience; laboratory activities in the undergraduate curriculumand evaluation of their effectiveness.Symplectic geometry is the mathematical framework for the studyof classical mechanics, and in particular Hamiltonian dynamicalsystems. Woodward will undertake a number of projects to developthe field and its role in the undergraduate and graduatecurriculum. He will seek to understand symmetries, especially ofinfinite dimension, in this setting. Possible applicationsoutside of mathematics are the development of new computationaltechniques in conformal field theory, an important topic inrecent high-energy theoretical physics, and improvement of theefficiency of wireless communication, via the construction ofbetter packings in the unitary group.