Projective Duality, Stability Conditions and Applications to Physics

The goal of this proposal is to develop the mathematical theory behind these new categorical structures and apply them to solve a broad variety of problems in physics and mathematics. We introduce a framework of geometrization of categories, in which geometrical structures appear out of categories in new ways. The newly developed categorical and geometric structures are used both in physics and in unexpected areas of mathematics. In addition to algebraic and symplectic geometry (a natural place to employ these theories), we are also proposing new applications to dynamical systems, differential equations, and number theory. The applications of higher categorical structures range from purely physics problems - e.g. incorporating fermions and global symmetries in higher dimensional quantum field theories (relevant for the study of the Fractional Quantum Hall Effect and its higher-dimensional analogs) to solving long-standing mathematical problems - e.g. proving that a generic four dimensional cubic is not rational.

The goal of this proposal is to develop the mathematical theory behind these new categorical structures and apply them to solve a broad variety of problems in physics and mathematics.We introduce a framework of geometrization of categories, in which geometrical structures appear out of categories in new ways. The newly developed categorical and geometric structures are used both in physics and in unexpected areas of mathematics. In addition to algebraic and symplectic geometry (a natural place to employ these theories), we are also proposing new applications to dynamical systems, differential equations, and number theory.The main physical source for these categorical developments is Mirror Symmetry, a physical duality between N = 2 superconformal field theories. In the 90's Maxim Kontsevich re-interpreted this concept from physics as a deep and ubiquitous mathematical duality, now known as Homological Mirror Symmetry (HMS). His 1994 lecture kick-started an explosion of activity in the mathematical community which lead to a remarkable synergy of diverse mathematical disciplines: symplectic geometry, algebraic geometry, and category theory. HMS is now the cornerstone of an immense field of active mathematical research.Numerous works by a range of authors have demonstrated the interaction of mirror symmetry and HMS with a wide range of new and subtle mathematical structures. We note in particular methods from Lagrangian intersection Floer theory, integrable systems and wall-crossing, derived and higher categories, and non-commutative Hodge structures, each of which are of immense independent mathematical interest, and each of which are intimately connected with HMS. Such connections have established HMS as a dominating force in modern geometry. The theory of Gaps and Spectra is one of the most spectacular achievements of HMS.We highlight here some of the notable recent results obtained. The report is organized as follows. We start with the obtained results. Then we present the dissemination efforts and broad impact of the project and report on the conference.The theory of linear systems is 2000 years old. Recently we have suggested a new read of this theory. We introduced the theory of categorical Kähler metrics. We develop further the theory of categorical linear systems.We take a new look at the categorical linear system applying the technique of sheaves of categories. We combine this technique with the theory of categorical linear systems and the theory of categorical Kähler metrics in order to build two parallels:1. A parallel with Donaldson theory of Kähler Einstein Metrics.2. A parallel with Donaldson theory of polynomial invariants.These directions have an immense impact on some classical questions of algebraic and Symplectic Geometry. The last two directions - developed in the last 3 years - are really ground breaking and open new venues of cutting edge research. We have produced high level postdoc and very well prepared graduate students - A. Noll, F. Haiden, G. Dimitrov. We have also worked with two visitors Y. Liu and L. Grama. The results we have obtained were recorded in several papers and 3 conferences allowed us to disseminate our new results. The above project has significant and broad output: 1. Deepening the connection with theoretical physics. 2. Establishing unexpected connection between category theory, complexity and dynamical systems. 3. Helping educate new generation of researchers through several. Our work has had a broad educational impact and is related to Physics.