Derived equivalences inspired by mirror symmetry

鲍里索夫的项目旨在进一步理解代数簇的派生类别，这些类别用于描述控制开弦传播的边界条件。他将验证双镜卡-丘完整的交叉环面品种的预期派生等价。他将致力于建设单位家庭的三角范畴之间的插值派生类别的不同crepant决议的复曲面奇点。鲍里索夫还将试图解决猜想，即具有相同的标准因子的双有理代数簇是等价的。弦理论是已知物理力的统一理论的领先物理候选者。尽管近年来弦理论取得了稳步的进展，但弦理论的数学方面还没有得到充分的理解。这种理解的缺乏是弦理论进一步发展的障碍，而弦理论的发展需要把它与真实的世界现象联系起来。鲍里索夫将有助于一个活跃的分支的数学研究，这是有关和启发弦理论，并将培养研究生在这一领域的研究。

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.