Spectral Invariants and Low-Dimensional Topology

Indiana University, Bloomington IN

Investigators

Abstract

DMS-0202148 Paul Kirk The PI proposes to study computations and applications of spectral invariants, in particular the eta-invariant of generalized Dirac operators, to problems in geometric topology and geometry. The focus will be to use analytic and differential geometric machinery to study spectral flow, the Atiyah-Patodi-Singer rho-invariant, and the $SU(3)$-gauge theoretic Casson invariant. The main goal is to develop usable cut-and-paste machinery for spectral invariants using the theory of boundary-value problems for Dirac operators. The PI's work in pure mathematics has as its goal the solution and clarification of geometric problems in 3 dimensions. The setting of 3 dimensions on the one hand is familiar to all of us since the universe we live in is 3-dimensional. But geometric problems in this area turn out to be very difficult for the odd reason that in such low dimensions there is "not enough room to move." The PI's approach to the area is to combine traditional 3 dimensional techniques with methods from other mathematical disciplines such as differential geometry and analysis.