Research Interests

Geometry, in particular symplectic/Poisson geometry and Lie theory, analysis, including determinants of operators, and mathematical physics.

My previous work included the study of a certain homogeneous Poisson structure on symmetric spaces discovered by Prof. Evens, of Notre Dame, and his collaborator, Dr. Lu, of the University of Hong Kong. In particular, I determined momentum maps for the actions of certain tori on the symplectic leaves. In concrete examples where the symmetric spaces are represented as coset spaces of matrix Lie groups, the components of these momentum maps turned out to be logarithms of ratios of determinants. Duistermaat-Heckman techniques from symplectic geometry can be applied using this data calculate geometric integrals of interest in representation theory. Portions of this work can be directly translated into the infinite dimensional geometric setting of loop spaces. It is there that subtle analytic issues arise and the determinants involved are determinants of operators. Computations of the analogous integrals in that setting should find application in the rigorous study of quantum field theory.

Selected Publications

• Compact symmetric spaces, triangular factorization, and Poisson geometry, Journal of Lie Theory, 18(2):273–294, 2008.
• with Pickrell, D., Homogeneous Poisson structures on symmetric spaces arXIV:0710.4484, to appear in International Mathematics Research Notices