Seminar talk, 6 October 2010

Speaker: Marina Prokhorova

Title: Spectral flow for a family of elliptic operators with local boundary conditions

Abstract:
Let X be a compact surface, E be a complex vector bundle over X, (A_t, L_t) be a 1-parameter family, such that A_t is a 1st order selfadjoint elliptic differential operator on E, L_t is a local selfadjoint elliptic boundary condition for A_t, and the pair (A_0, L_0) goes to (A_1, L_1) under an unitary automorphism of the bundle E. The spectrum of the operator (A_t, L_t) is discrete, real, and continuously depends on t. As t changes from 0 to 1, the spectrum shifts by an integer number of points, since the initial and final operators are isospectral. This number is called the spectral flow of the family of operators (A_t, L_t). I will tell how to compute the spectral flow in this situation.