Category theory is an algebraic approach that formalizes the study of mathematical structures. This project lays the foundations of classical geometric methods in category theory. Today, in the second decade of the 21st century, modern geometry and theoretical physics are more intertwined than ever before. The convergence of ideas from mathematics and physics is accelerating at the same time as elementary particle physics is on the cusp of a profound revolution to be brought about by the new experimental results coming out of the Large Hadron Collider (LHC). At the same time, a lot of mathematical work remains to be done to provide a suitable framework for the new physical theories that are being proposed. The geometric objects investigated in this project are the foundations for such a framework: homological mirror symmetry is the mathematical realization of dualities and higher categories, the analogues of classical manifolds, are the mathematical foundation for quantum field theories. These new flavors of geometry on which this project is based will continue to play a fundamental role in the future development of theoretical physics.
The proposed approach is based on the pioneering works by Seidel, Ein, Lazarsfeld, Mustata, Nakamaye, Popa, and Budur. The PIs will go further and conjecture that the categorical multiplier ideal sheaf is related to the Orlov spectrum of the category. Developing K-calculus and making it rigorous the PIs will break new ground in studying classical questions in Algebraic Geometry, including questions of rationality of projective varieties. In particular the PIs plan to consider some more than hundred years old questions about nonrationality of conic bundles and four dimensional cubics. The applications go beyond the scope of Algebraic Geometry. Classical questions in Sympletcic Geometry will be studied as well - using invariants of Fukaya category one can try to distinguish symplectic manifolds with the same Seiberg - Witten invariants. The approach connects K-calculus with so called tasting configurations used in the study of the existence of Kahler-Einstein metrics. An intriguing question is that of finding a connection between Orlov spectra and the existence of Kahler-Einstein metrics. Another direction of the project is the investigation of the connection of K-calculus with physics, for which a starting point is the interpretation of monodromy data of the K - calculus as limited stability conditions.