Derived Categories, Noncommutative Orders, and Other Topics

拟议的研究是在代数几何领域，与弦理论有一些联系。代数几何是数学的一个分支，研究由多项式方程和相关数学结构定义的几何对象。经典地，人们把这样的几何对象(称为代数簇)与其上的代数函数集联系在一起，形成一个交换环(即，函数可以被加法和乘法)。现代研究涉及与代数簇相关的更复杂的代数结构，例如凝聚层范畴(范畴的概念是结合环的概念的推广)。该项目的一部分是建立同调镜像对称猜想的一些情况，该猜想识别在两个看似无关的上下文中出现在几何中的范畴。该项目的另一部分旨在为弦理论中使用超黎曼曲面(通常曲面的推广)的某些方面提供严格的数学基础。这个项目为本科生和研究生提供了研究培训的机会。更具体地说，项目的第一部分是关于被穿孔球体的对称幂的同调镜像对称性。其目的是确定某些代数簇上部分包裹Fukaya范畴的凝聚层派生范畴的范畴分解。这可能有助于找到Ozsvath-Szabo绝对纽结不变量的新构造。第二部分是将Hirzebruch-Riemann-Roch公式推广到非仿射簇和堆栈上的矩阵分解范畴。PI还希望使用矩阵分解的范畴来寻找G等变Gromov-Witten理论的Landau-Ginzburg对应。项目的第三部分是实现结合型Yang-Baxter方程在结点三次曲面上的非对易阶数三角解。第四部分是关于稳定超曲线的几何问题。PI建议理解通过稳定超曲线紧凑边界附近的超曲线的模的Berezian的Mumford同构的模拟的极点，并研究在超曲线的模空间上的积分中出现的一些问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The proposed research is in the field of algebraic geometry with some connections to string theory. Algebraic geometry is a branch of mathematics studying geometric objects defined by polynomial equations and related mathematical structures. Classically one associates with such geometric objects (called algebraic varieties) the set of algebraic functions on them which forms a commutative ring (i.e., functions can be added and multiplied). Modern research involves more sophisticated algebraic structures associated with algebraic varieties, such as the category of coherent sheaves (the notion of a category is a generalization of that of an associative ring). One part of the project is to establish some cases of the homological mirror symmetry conjecture which identifies categories appearing in geometry in two seemingly unrelated contexts. Another part of the project aims to give a rigorous mathematical foundation to some aspects of the use of super Riemann surfaces (a generalization of the usual surfaces) in string theory. This project provides research training opportunities for undergraduate and graduate students.More specifically, the first part of the project is on homological mirror symmetry for symmetric powers of punctured spheres. The goal is to identify categorical resolutions of derived categories of coherent sheaves on certain algebraic varieties with partially wrapped Fukaya categories of the symmetric powers of punctured spheres. This may help to find a new construction of Ozsvath-Szabo's categorical knot invariant. The second part is to work out a generalization of the Hirzebruch-Riemann-Roch formula to the categories of matrix factorizations over non-affine varieties and stacks. The PI also would like to use categories of matrix factorizations to find a Landau-Ginzburg counterpart of the G-equivariant Gromov-Witten theory. The third part of the project is to realize trigonometric solutions of the associative Yang-Baxter equation in terms of noncommutative orders over nodal cubics. The fourth part is related to the geometry of stable supercurves. The PI proposes to understand the poles of the analog of Mumford's isomorphism for the Berezinian of the moduli of supercurves near the boundary of the compactification by stable supercurves and to study some problems arising in integration over the moduli space of supercurves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.