Grad Student/Postdoc Seminar 
March 26, 2004:  Jean Steiner, CIMS 

Determinants, traces, and 'hearing' the shape of a surface

In  linear algebra the determinant and the trace are useful tools in
understanding the behavior of an nxn matrix, a linear operator whose
domain is  a finite dimensional space, R^n.  In geometry, the
determinant and the trace also turn out to be valuable tools:  on a
given surface such as the sphere or the torus,  the linear operator
is the Laplacian operator whose domain is  an infinite dimensional
space of functions.  Since the Laplacian has infinitely many
(unbounded!) eigenvalues, we will begin by discussing  how one can
make sense of quantities such as the determinant and the trace
through a regularization procedure.  Then we will highlight some of
the rich geometrical information that is `heard' by the determinant
and the trace.  In keeping with the aim of the seminar, this talk is
intended to just give a flavor of a few tools and problems in
spectral geometry.