Noncommutative Geometry

该计划涉及代数K理论，同调代数，重点是循环和Hochschild同调，以及无穷小几何等领域的调查。具体而言，正在构建和研究算子的不变量：奇异迹，高指数不变量和调节器，以及它们对狄利克雷级数理论的影响及其在临界线附近的行为。微分算子和符号代数的结构、同调和不变量是无穷小几何研究的重点。最后，非交换几何中与特殊导子和奇异链同伦等价有关的新现象取代了德拉姆理论的Koszul解析方法，是同调代数研究的主题。 本计划中的项目涉及非交换几何、同调代数、泛函和全局分析的基本主题。他们提供了新的方法，一些经典科目在其他领域的数学和数学物理，特别是代数几何-包括其算术方面，奇点理论，分形几何，分析量子流形。

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.