Differential graded categories and their applications in geometry

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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.