Mathematical Sciences: Non-Commutative Differential Geometry of Deformations of Commutative Rings: Operations Index Theorems and Characteristic Classes

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9623051 Tsygan This grant supports the research of Professor B. Tsygan to work on various problems in the deformation theory of Poisson manifolds. In particular he will work on algebraic index theorems, the homological algebra of characteristic classes and quantum cohomology and applications of structure theorems for deformations to both Riemann-Roch and the Verlinde formulas. This is research that starts out in the field of algebra but moves beyond that to touch both geometry and theoretical physics. The reasons for these connections are that algebra can be though of as the study of symmetry in the abstract. As such algebra has direct applications to areas of physics and chemistry. In particular the modern theory of gauge fields in physics uses this area of algebra extensively and this project will study some of these connections at length.