Topology, Noncommutative Geometry, and Applications

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Professor Jonathan Rosenberg will study problems in classical and noncommutative topology, as well as various applications, especially to differential geometry and mathematical physics. One main focus of the proposal will be the application of noncommutative geometry to mathematical physics, especially string theory. Noncommutative sigma-models will be analyzed and compared with their classical counterparts. This will require development of some aspects of a new theory of noncommutative nonlinear elliptic partial differential equations. Other topics will include the classification of metrics of positive scalar curvature on manifolds, the purely algebraic K-theory of operator algebras, and the use of invariants coming from C*-algebras (especially Kasparov?s KK-theory) to study the geometry and topology of manifolds. For example, the K-homology classes coming from the classical elliptic operators, such as the signature operator and the Dolbeault operator, will be computed, and their invariance and rigidity properties will be determined. As a result, we expect to understand better the links between topological and analytic approaches to geometry of manifolds and singular spaces.Many physical theories, such as general relativity, are formulated in terms of geometry and partial differential equations. However, the principles of quantum mechanics require studying space-time ``geometries''in which the coordinate functions do not commute with one another. One main focus of this project will be the reformulation of some of the partial differential equations of mathematical physics in the setting of such noncommutative geometries. We will develop tools for studying these noncommutative equations and will compare the noncommutative geometries with their classical counterparts. This should advance the language for formulating quantum theories of gravity.Professor Rosenberg will also train graduate students and advanced undergraduate students in algebra, analysis, geometry, topology, and mathematical physics, and will also work toward integration of mathematical software into the undergraduate mathematics curriculum.