Topology of Manifolds, C*-Algebras, and Applications

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Proposed abstract:Professor Jonathan Rosenberg will study the topology and geometry of manifolds, as well as C*-algebraic index theory, and various applications. The applications will include a number of topics relevant to modern physics,such as the classification of manifolds of positive scalar curvature,relevant to the problem of understanding the possibilities for thelarge-scale geometry of space-time, and questions relevant to dualities between different space-time topologies in string theory. Professor Rosenberg is closely involved in the supervision and improvement of the mathematics graduate programs at the University of Maryland. He will 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.In more detail, one main focus of the proposal will be the classification of manifolds via their Yamabe invariants, and the classification of metrics of positive scalar curvature, problems which involve a quite subtle blend of differential topology, differential geometry, and analysis. Another focus will be applications of topology and noncommutative geometry to mathematical physics, especially to T-duality in string theory. Other topics will include the use of invariants coming from C*-algebras (especially Kasparov's KK-theory) to study the geometry and topology of manifolds. For example, the KK-classes coming from the classical elliptic operators, such as the signature operator and the Dolbeault operator, will be intensively studied. These studies will deepen the link between topological and analytic approaches to geometry of manifolds and singular spaces. In addition, Professor Rosenberg will study the applications of K-theory to the study of C*-algebras, and various related problems on algebraic K-theory, especially concerning algebras of operators or algebras coming from quantization of geometrical systems.