12 Marzo, 2012 17:00

Seminario Matematico e Fisico di Milano

REGULARIZED DETERMINANTS AND CONFORMALLY INVARIANT OPERATORS

MATTHEW GURSKY, University of Notre Dame

Università di Milano, Dipartimento di Matematica, Via Saldini

In this talk I will give an overview of some geometric and analytic issues related to the regularized determinant of an elliptic operator.
I will begin with the work of Osgood-Phillips-Sarnak on the determinant of the laplacian for surfaces, which has its origins in a formula of Polyakov, and explain the connection to the uniformization theorem and the Ricci flow. In four dimensions, the starting point is a beautiful but daunting formula of Branson-Orsted for conformal variations of the determinant. I will explain the connection of this formula to mathematical physics and conformal geometry, then discuss some of it variational properties. I will end with a question posed by Connes about the determinant of the Paneitz operator, and some work in progress.