Derived equivalences inspired by mirror symmetry

鲍里索夫的项目旨在加深对代数簇派生范畴的理解，代数簇用于描述控制开弦传播的边界条件。他将验证环簇中双镜Calabi-Yau完全交的预期导出等价性。他将致力于构建三角化范畴的平面族，这些平坦族在环面奇点的不同初始分辨率的派生范畴之间进行内插。鲍里索夫还试图解决这样一个猜想，即具有相同正则因子的二元代数簇是等价的。弦理论是已知物理力的统一理论的主要物理候选者。尽管近年来取得了稳步的进展，但弦理论的数学方面还没有得到充分的理解。这种缺乏理解是弦理论进一步发展的障碍，弦理论是将弦理论与现实世界现象联系起来所必需的。鲍里索夫将为与弦理论相关并受到弦理论启发的数学研究的一个活跃分支做出贡献，并将培养这一研究领域的研究生。

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.