Noncommutative analysis and geometry in interaction with quantum physics

The interaction between theoretical models in quantum physics and mathematics has led to some of the most exciting mathematical advances of the last 30 years. Yet many puzzles remain in both fields. In this proposal we study mathematical questions arising in geometry by borrowing ideas from quantum mechanical scattering theory, supersymmetric quantum models and quantum statistical mechanics. In each case we present novel applications of these ideas to the problem of constructing invariants of geometric structures. These structures can be both classical, in the form of higher dimensional surfaces, or noncommutative algebras with origins in physical models of quantum phenomena. We will attack some prominent unresolved mathematical questions.