Non Commutative Geometry, Microlocal Analysis, and Symplectic Geometry

提出的研究旨在回答非交换几何，辛几何和微局部分析的几个问题。特别是，我将继续我对指数定理的研究。我将致力于将Atiyah-Singer指标定理从伪微分推广到傅里叶积分算子，并将cones和Moscovici的新指标定理与变形量化的一般指标定理联系起来。我还将研究一个更一般的特征类理论。这将导致，特别地，将chen - simons类从局部系统推广到任意d模块。反过来，这将有助于更好地理解和推广上同调行列式的黎曼-洛克定理。我的另一个研究方向是辛流形上函数的变形代数上的模理论。首先，我打算定义一个具有一些附加结构的新模块类别；它应该与深谷类有关。前者应被视为前者的局部极限。下一步将是使这个类别非本地，因此与深谷类别更密切相关。我的研究涉及几个相关的主题。一种是非交换几何，它的大部分是由通常的微积分组成的，但是以这样一种方式进行，即恒等式xy=yx不再有效。我们把这种情况下的代数系统称为非交换空间。另一个是变形量化，即研究非交换空间的例子和结构，而不是研究非交换微积分的规律和规则。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稍微改变它来获得这些非交换空间。（之所以使用“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）。我研究的另一个方向是指标论，这是一门将方程系统（微分、积分等）的解的数量与底层空间的拓扑复杂性联系起来的学科。它与非交换几何密切相关，因为你可以应用于函数的两个运算，乘以x和求导，是不能交换的。

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geometry, much of it consists of the usual calculus, but carried out in such a way that the identity xy=yx is no longer valid.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.