Spectral Invariants and Low-Dimensional Topology

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DMS-0202148Paul KirkThe PI proposes to study computations and applications ofspectral invariants, inparticular the eta-invariant of generalized Dirac operators, toproblems in geometric topology and geometry. The focus will be touse analytic and differential geometric machinery to study spectralflow, 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 usingthe theory of boundary-value problems for Dirac operators.The PI's work in pure mathematics has as its goal the solution andclarification of geometric problems in 3 dimensions. The setting of3 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 withmethods from other mathematical disciplines such as differential geometry and analysis.