RESEARCH STATEMENT

My research lies at the intersection of Lie theory and Poisson geometry. The funda-mental object of study is a Poisson Lie group, a Lie group G together with a Poisson structure (a bivector field satisfying the Jacobi identity, in a certain sense) that is com-patible with the group multiplication on G: the map G × G → G: (g1, g2) 7 → g1g2 is a Poisson map. Thus, a Poisson Lie group is a space that is endowed with three structures, a smooth structure, a group structure, and a Poisson structure, all of which are somehow compatible. Poisson structures can be thought of as symplectic structures for which degeneracy is allowed. Generalizing from symplectic to Poisson manifolds yields a framework in which a broader class of problems can be modeled, including many problems in physics. Since Lie groups and Lie group actions also play an important role in both physics and various fields of mathematics, it is natural to consider Lie groups with compatible Poisson structures as well as actions of Lie groups on Poisson manifolds. Furthermore, actions of Poisson Lie groups on Poisson manifolds can be thought of as generalizations of Lie group actions on symplectic manifolds, and the notion of a momentum map for a symplectic action ca