Factorizations, Projective Duality, and Cycles

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Traditionally, algebra and geometry are studied together by thinking of solutions to equations as shapes. For example circles, lines, and parabolas are defined by polynomial equations. This is the basis of Algebraic Geometry. This research program adds a third ingredient, inspired by the great attempt in High-Energy Theoretical Physics to unify Einstein's Theory of Gravity with Quantum Mechanics. Einstein's Theory of Gravity is incredibly accurate when predicting the movement of planets, comets, and stars while the Standard Model of Quantum Mechanics has been verified with amazing precision at the microscopic level. However, at high energy these two theories are incapable. One proposal to unify these theories is that the universe itself is made up of tiny little vibrating strings (as opposed to infinitesimal point-like particles). While this has not been experimentally verified in the real world, it it has incredible implications towards our understanding of geometry. An important aspect of Einstein's Theory of Gravity is that he viewed space and time not as separate entities but as one continuous "4-dimensional" geometry - spacetime. Now, let's treat spacetime like something we all deal with daily: H2O. As H20 changes temperature it transitions through various phases: water, ice, steam. Physicists predict that spacetime can also undergo "phase-transitions" which change aspects of its geometry. Some of these changes are actually well understood through birational geometry, one of the most classical aspects of Algebraic Geometry. On the other hand, physicists often incorporate additional data into their study of spacetime for example in so-called Landau-Ginzburg models. In this proposal, we concentrate on the implications in Algebraic Geometry of predictions from High-Energy Physics such as a mathematical interpretation of "phase-change" for Landau-Ginzburg models. Specifically, the purpose of the project is to create and implement rigorous mathematical machinery for derived categories of Landau-Ginzburg models, algebraic cycles, and Homological Mirror Symmetry. The main novelty is that, by using physical interpretations of space and time, we can produce new and unexpected results about algebraic equations and geometric shapes.