Noncommutative Geometry

Wodzicki将继续他对代数K-理论、算子代数的循环同调和Hochschild同调以及算子理想的研究；他将研究算子不变量(例如，高指数不变量和调节子)的构造。他还将学习奇异迹和由此产生的不变量；经典几何、分数几何和非交换几何的应用；狄里克莱级数理论的分支及其在狄里克莱级数临界线附近的行为。他还打算研究非对易几何中与特殊导子和奇异链同伦等价有关的新现象，这些现象取代了de Rham理论的Koszul分解方法，特别是对可能应用于量子群、量子齐次空间和非对易流形的应用感兴趣。除了上一段提到的领域外，代数和解析几何、奇点理论、分数几何和数学物理是拟议研究影响最大的领域。

AbstractWodzickiThe PI will continue his investigation of algebraic K-theory, cyclic and Hochschild homology of operator algebras and operator ideals; he will be studying constructions of invariants of operators (e.g., higher index invariants and regulators). He will also study exotic traces and resulting invariants; the applications to Classical as well as Fractal, and Noncommutative Geometry; ramifications for the theory of Dirichlet series and their behavior in the vicinity of the critical line of Dirichlet series. He also intends to investigate new phenomena in Noncommutative Geometry related to special derivations and exotic chain homotopy equivalences that replace Koszul resolution approach to de Rham theory, with a particular interest in possible applications to quantum groups, quantum homogeneous spaces and noncommutative manifolds.Besides the areas mentioned in the previous paragraph, Algebraic andAnalytic Geometry, Singularity Theory, Fractal Geometry and MathematicalPhysics are the areas where the impact of the proposed investigations will be most significant.