Mathematical Sciences: Presidential Young Investigator Award

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This Presidential Young Investigator award will support the research of Professor Wodzicki in the area of noncommutative differential geometry. The dominant themes in this research will be a noncommutative Riemann-Roch theorem, higher index invariants, connections with arithmetic algebraic geometry via noncommutative differential geometry in positive characteristic, and quantization of homological algebras in a direction parallel to quantum groups. Differential geometry can be thought of as calculus on manifolds. These are surfaces and their higher dimensional analogues. The algebra of differentiation operators on the manifold can be further abstracted until there is no manifold left but just the algebra. This is the far reaching generalization of differential geometry called noncommutative differential geometry. The area draws on and nourishes all the major fields of mathematics: algebra, geometry and analysis.