Noncommutative Geometry

这项建议涉及的研究领域包括代数K-理论，同调代数，重点是循环和Hochschild同调，以及无穷小几何。具体地说，正在构造和研究算子的不变量：奇异迹、高指数不变量和调节器，以及它们对Dirichlet级数理论的影响和它们在临界线附近的行为。与微分算子和符号的代数有关的结构、同调和不变量是《无穷小几何》的研究重点。最后，非对易几何中与特殊导子有关的新现象和取代de Rham理论的Koszul分解方法的奇异链同伦等价是同调代数研究的主题。本提案中的项目涉及非对易几何、同调代数、泛函和整体分析等基本主题。它们为数学和数学物理的其他领域中的一些经典学科提供了新的方法，特别是代数几何--包括它的算术方面、奇点理论、分形几何、量子流形分析。

The proposal involves investigations in the areas of algebraic K-theory, Homological Algebra, with an emphasis on Cyclic and Hochschild Homology, and Infinitesimal Geometry. Specifically, invariants of operators are being constructed and studied: exotic traces, higher index invariants and regulators, as well as their ramifications for the theory of Dirichlet series and their behavior in the vicinity of the critical line. The structure, homology, and invariants associated to the algebras of differential operators and symbols are the focus of the investigation in Infinitesimal Geometry. Finally, new phenomena in Noncommutative Geometry related to special derivations and exotic chain homotopy equivalences that replace Koszul resolution approach to de Rham theory are the subject of the investigation in Homological Algebra. The projects in this proposal are concerned with fundamental topics in Noncommutative Geometry, Homological Algebra, Functional and Global Analysis. They offer novel approaches to some classical subjects in other areas of Mathematics and Mathematical Physics, in particular, Algebraic Geometry -- including its Arithmetic aspects, Singularity Theory, Fractal Geometry, Analysis on Quantum Manifolds.