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.

No additional funding sources recorded.