Index Theory and Spectral Flow for Non-unital Spectral Triples and Twisted Spectral Triples

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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 mechanics in the form of Alain Conne's so-called "noncommutative" geometry: a synthesis of algebraic and geometric ideas. We present novel applications of these ideas to the problem of constructing invariants of both commutative and noncommutative geometric structures. Examples include higher dimensional surfaces or topological spaces, and also noncommutative algebras with origins in physical models of quantum phenomena. We will attack some prominent unresolved mathematical questions.