Topology, Noncommutative Geometry, and Mathematical Physics

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This research project explores questions at the interface of geometry in mathematics and string theory in physics. The project aims to investigate the structure of physical theories by using methods from the mathematical fields of non-commutative geometry and K-theory, and in turn, to utilize ideas from physics to study new phenomena in geometry and topology. The results are expected to advance several research areas in mathematics and physics. Graduate and undergraduate students will receive research training through involvement in the interdisciplinary project, and the investigator will continue active leadership in the K-theory community.The three primary areas of focus for the project are the relationship between topological K-theory and string theory, the geometry of noncommutative manifolds, and the study of metrics of positive scalar curvature and the Yamabe invariant. In the first area, the project will study how K-theory constrains and suggests dualities in string theory, F-theory, and M-theory, and why K-theory isomorphisms associated with T-dualities seem to be related to the Baum-Connes conjecture. In the second area, the project aims to understand the geometry of metrics on the noncommutative 2-torus that are not conformally flat, and to prove a noncommutative Gauss-Bonnet theorem for such metrics. In the last area, the project will attempt to establish what curvature positivity conditions guarantee vanishing of terms in the Witten genus, and to better understand the topology of the space of positive scalar curvature metrics on a manifold.