Research Interests

My research is in Lie theory , especially its connections to representation theory and loop groups . Loop groups are infinite-dimensional groups whose elements are smooth maps from the circle ("loops") into a finite-dimensional Lie group (think of a matrix group, like the group of unitary matrices), and where multiplication is pointwise along the circle. Thus, a loop group can be thought of as a space of matrices whose entries depend on a periodic parameter, theta, or a complex parameter, z, which is restricted to lie on the circle.

I am studying a new kind of factorization for these matrices. This factorization allows us to write an element of a loop group as a product of infinitely many simple elements. Each simple element is associated with a power of z, and a complex coefficient, so this factorization is like a nonabelian Fourier transform. Just as with the regular Fourier transform, there are relationships between the decay properties of the coefficients and the smoothness of the resulting loops. There are also promising connections to measure theory and Poisson geometry, and conjectured applications in theoretical physics. A more detailed description of this research, as well a some of my other interests, can be found in my research statement .