Topology, Noncommutative Geometry, and Mathematical Physics

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This research program combines the areas of topology, noncommutative geometry, and mathematical physics. One focus of this proposal will be on applications of topology and noncommutative geometry to duality in string theory, in order to understand topological constraints on dualities between string theories on different spacetime backgrounds, and also to understand situations where duality leads to a field theory on a noncommutative space. Another major area of investigation will be the use of invariants from noncommutative geometry and index theory for studying the geometry and topology of manifolds and of manifold-like spaces, such as manifolds with singularities. Rosenberg will also use methods of geometric analysis and differential topology to study the topology of the space of Riemannian metrics of positive scalar curvature on a manifold, in both low and high dimensions. Finally, the PI will continue preparation of course materials for the use of mathematical software in sophomore-level science/engineering courses in multivariable calculus and differential equations.The interaction between mathematics (especially geometry) and physics has been a long and fruitful one for both disciplines. This project will advance that interaction, by relying on physics to suggest interesting topics of study in geometry, and using our understanding of geometry to put restrictions on fundamental physical theories. Broader impacts of this project will include (a) mentoring of graduate and undergraduate students, (b) development of materials and courses for the training of undergraduates, graduate students and postdocs, and (c) dissemination of research in geometry and mathematical physics to a wide audience.