Mathematical Aspects of IIA-IIB Models of Superstring Theory

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Borisov will study in detail the IIA-IIB superstring theoriesthat correspond to Calabi-Yau hypersurfaces in toric varieties. The goal is to provide geometric meaning to the combinatorialconstruction of N=2 vertex algebras associated with such theories.In another part of the project, Borisov will try to prove the derived equivalence conjecture for K-equivalent algebraic varieties.His approach aims at defining a triangulated category ofany log-terminal pair, which would extend the notion of the derived category of a variety to this more general context.Finally, Borisov will strive to settle a puzzling discrepancy betweenthe numbers of parameters of deformation of small quantum cohomologyof a DM stack and of those of a crepant resolution of itscoarse moduli space.Generally, Borisov will try to build mathematical tools necessary for further development of string theory. In the particular flavor of string theory that Borisov's project is related to, stringsare expected to propagate on a "small" space of dimension six withvery peculiar properties. Borisov's approach may generalize this smallspace to other mathematical data, which may potentially increase the flexibility of the theory.