Toric Geometry and Mirror Symmetry

鲍里索夫的项目由三个独立但相关的部分组成。在第一部分中，他将试图获得一个更明确的描述，即对环形戈伦斯坦奇点的叠层蠕变分辨率的派生范畴。Borisov的目标是找到这些类别族的自包含代数描述，这有可能简化开放超弦理论的IIB型模型的计算。该项目的第二部分试图证明或反驳King关于Fano toric品种和堆栈存在倾斜束的猜想。Borisov项目的最后一部分旨在证明，在环面变种中，网状超曲面的Calabi-Yau完全交集只有有限多个族。鲍里索夫的研究涉及弦理论的数学方面。他的专业领域是环面几何，这是一个研究被称为环面变异的高维空间的数学领域。环面几何是在20世纪70年代发展起来的，用于纯数学目的，作为构造空间显式示例的工具。20世纪90年代初，人们认识到环面变异在超弦理论中起着重要的作用，这一理论开始得到重视，并得到迅速发展。鲍里索夫希望将他在环面几何方面的专业知识应用于超弦理论中出现的几个数学问题。特别是，他希望确定目前已知的超弦可能运动的目标空间的结构最多可以产生有限（尽管非常大）数量的不同族。

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.