Differential graded categories and their applications in geometry

该项目的目标是研究和开发微分分级类别（DG）装置在几何和数学物理的几个问题中的应用。 首先，PI 希望开发 Kashiwara-Schapira 的可构造层微局部理论的 DG 版本，然后，使用该装置，在给定辛流形的情况下，他希望构造一个 dg 范畴并将其与 Fukaya 范畴进行比较。PI 还希望继续他在数学上严格的泊松西格玛模型、非交换微积分（与他的合作者）和理论方面的工作。 n 类别。微分分级 (DG) 类别在代数几何和数学物理的各种问题中发挥着重要作用。该项目的目标是进一步开发DG类别的一些应用。其中一个应用程序旨在提供著名的 Fukaya 类别的替代结构，这种结构在精神上应该更加几何化和透明。 PI 还希望继续研究他之前研究的几个主题，例如量子场论的数学严格模型、非交换微积分（与他的合作者）以及 DG 类别装置的进一步开发。

The goal of this project is to study and develop applications of the apparatus of differential graded categories (DG) to several problems of geometry and mathematical physics. First, the PI would like to develop a DG version of the microlocal theory of constructible sheafs due to Kashiwara-Schapira and then, using this apparatus, given a symplectic manifold, he would like to construct a dg-category and to compare it with the Fukaya category.The PI would also like to continue his work on mathematically rigorous Poisson sigma-model, non-commutative calculus (with his collaborators), and theory of n-categories.Differential graded (DG) categories play an important role in a variety of problems in algebraic geometry and mathematical physics. The goal of this project is to further develop some applications of DG categories. One of these applications is aimed at providing an alternative construction of the celebrated Fukaya category, this construction is supposed to be more geometric and transparent in spirit. The PI would also like to continue on several topics from his previous research such as mathematically rigorous models of quantum field theory, non-commutative calculus (with his collaborators), and further development of the apparatus of DG categories.