Mathematical Aspects of IIA-IIB Models of Superstring Theory

鲍里索夫将详细研究IIA-IIB超弦理论，对应于复曲面簇中的Calabi-Yau超曲面。我们的目标是为与这些理论相关的N=2顶点代数的组合构造提供几何意义。在项目的另一部分，鲍里索夫将试图证明K-等价代数簇的导出等价猜想。他的方法旨在定义任何对数终结对的三角范畴，这将把簇的导出范畴的概念扩展到更一般的背景。最后，鲍里索夫将致力于解决DM栈的小量子余同调的变形参数数与其粗糙模空间的可分辨变形参数数之间令人困惑的差异。总的来说，鲍里索夫将试图建立进一步发展弦理论所必需的数学工具。在鲍里索夫的项目所涉及的弦理论的特殊风格中，弦被期望在具有非常特殊性质的六维“小”空间中传播。鲍里索夫的方法可以将这个小空间推广到其他数学数据，这可能会增加理论的灵活性。

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.