Toric Geometry and Mirror Symmetry

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Borisov's project consists of three separate but related parts. In the first part, he will attempt to obtain a more explicit description of the derived categories of the stacky crepant resolutions of toric Gorenstein singularities. Borisov's goal is to find a self-contained algebraic description of the families of these categories, which has the potential to simplify the calculations in type IIB models of open superstring theory. The second part of the project attempts to prove or disprove a conjecture of King on the existence of tilting bundles for Fano toric varieties and stacks. The last part of Borisov's project aims to show that there are only finitely many families of Calabi-Yau complete intersections of nef hypersurfaces in toric varieties.Borisov's research concerns mathematical aspects of string theory. His particular area of expertise is toric geometry which is an area of mathematics that studies higher-dimensional spaces known as toric varieties. Toric geometry was developed in 1970s for purely mathematical purposes as a tool for constructing explicit examples of spaces. It came to prominence in the early 1990s when it was realized that toric varieties play an important role in superstring theory, and it has been developing rapidly ever since. Borisov hopes to apply his expertise in toric geometry to several mathematical problems that appear in superstring theory. In particular, he hopes to establish that the currently known constructions of possible target spaces in which superstrings are supposed to move can produce at most a finite (albeit very large) number of different families.