Mathematical Sciences: Presidential Young Investigator Award

总统青年研究员奖将支持Wodzicki教授在非交换微分几何领域的研究。本研究的主要主题将是非交换黎曼-洛克定理，高指数不变量，通过非交换微分几何在正特征上与算术代数几何的联系，以及平行于量子群方向上同调代数的量化。微分几何可以看作是流形上的微积分。这些是表面和它们的高维类似物。流形上的微分算子代数可以进一步抽象，直到流形只剩下代数。这是微分几何的深远推广称为非交换微分几何。该领域吸收并滋养了数学的所有主要领域：代数、几何和分析。

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.