Borisov's project aims to further the understanding of derived categories of algebraic varieties, which are used to describe boundary conditions that govern the propagation of open strings. He will verify the expected derived equivalence of double mirror Calabi-Yau complete intersections in toric varieties. He will work to construct flat families of triangulated categories that interpolate between derived categories of different crepant resolutions of a toric singularity. Borisov will also attempt to settle the conjecture that states that birational algebraic varieties with the same canonical divisor are derived equivalent.
String theory is the leading physical candidate for the unified theory of the known physical forces. Despite steady progress in recent years, mathematical aspects of string theory are not yet adequately understood. This dearth of understanding is an obstacle to further development of string theory which is needed to connect it with the real world phenomena. Borisov will contribute to an active branch of mathematics research which is related to and is inspired by string theory, and will train a graduate student in this area of study.
