Non Commutative Geometry, Microlocal Analysis, and Symplectic Geometry

建议的研究是为了回答几个问题的非交换几何，辛几何和微局部分析。特别是，我将继续我的研究指标定理。我将致力于推广的Atiyah-Singer指标定理从伪微分傅立叶积分算子和连接的新指标定理的Connes和Moscovici的一般指标定理的变形量化。我还将研究更一般的特征类理论。这应该导致，特别是，一个推广的陈-西蒙斯类从本地系统任意D-模。这反过来又会导致更好地理解和推广黎曼-罗克定理的决定因素的上同调。我的另一个研究方向将是辛流形上的变形函数代数上的模理论。首先，我打算定义一个新的带有一些附加结构的模块类别;它应该与福谷类别相关。前者应视为前者的局部界限。下一步将是使这个范畴非本地化，因此与福谷范畴的关系更加密切。一个是非交换几何，其中大部分由通常的微积分组成，但以这样一种方式进行，即恒等式xy=yx不再有效。我们把这种情况称为非交换空间的代数系统。另一个是形变量子化，即研究非对易空间的例子和结构，而不是非对易微积分的定律和规则。变形这个词指的是这样一个事实，即人们通过取一个通常的（弯曲的，高维的）空间并在非交换方向上改变它来获得这些非交换空间。(The使用量子化这个词是因为从经典力学到量子力学的过渡是主要的例子）。我的另一个研究方向是指数理论，这是一个学科联系方程组（微分，积分等）的解决方案的数量。到底层空间的拓扑复杂性。它与非对易几何有很深的联系，因为你可以应用于函数的两个运算，乘以x和求导，不对易。

The proposed research is intended to answer several questions of noncommutative geometry, symplectic geometry, and microlocal analysis. In particular, I will continue my research on index theorems. I will work on a generalization of the Atiyah-Singer index theorem from pseudo-differential to Fourier integral operators and on a connection of the new index theorems of Connes and Moscovici to the general index theorem for deformation quantizations. I will also work on a more general theory of characteristic classes. This should lead, in particular, to a generalization of the Chern-Simons classes from local systems to arbitrary D-modules. This, in turn, should lead to better understanding and generalization of Riemann-Roch theorems for determinants of cohomology. Another direction of my research will be the theory of modules over deformed algebra of functions on a symplectic manifold. First, I intend to define a new category of modules with some additional structure; it should be related to the Fukaya category. The former should be regarded as a local limit of the former. Next step would be to make this category non-local and therefore even more closely related to the Fukaya category.My research deals with several related topics. One is noncommutative geometry, much of it consists of the usual calculus, but carried out in such a way that the identity xy=yx is no longer valid. We refer to algebraic systems in which this is the case as noncommutative spaces. The other is deformation quantization, namely the study of the examples and structure of noncommutative spaces, rather than of laws and rules of noncommutative calculus. The word deformation refers to the fact that one obtains these noncommutative spaces by taking a usual (curved, higher-dimensional) space and changing it a little bit in noncommutative direction. (The word quantization is used because the passage from the classical to the quantum mechanics is the principal example). Yet another direction of my research is index theory, which is a discipline linking the number of solutions of systems of equations (differential, integral, etc.) to the topological complexity of underlying spaces. It is deeply related to noncommutative geometry because two operations that you can apply to a function, multiplication by x and derivation, do not commute.