Derived Categories, Noncommutative Orders, and Other Topics

本研究方向为代数几何领域，与弦理论有一定的联系。代数几何是研究由多项式方程和相关数学结构定义的几何对象的数学分支。经典地，人们将这样的几何对象（称为代数变量）与在其上形成交换环的代数函数集联系起来（即函数可以相加和相乘）。现代研究涉及与代数变异相关的更复杂的代数结构，例如相干束的范畴（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一些同调镜像对称猜想的例子，该猜想可以识别在两种看似无关的背景下出现在几何中的类别。该项目的另一部分旨在为超黎曼曲面（通常曲面的推广）在弦理论中的某些方面的应用提供严格的数学基础。本项目为本科生和研究生提供研究训练机会。更具体地说，项目的第一部分是关于穿孔球体对称幂的同调镜像对称。目的是确定某些代数变体上相干束的派生范畴的范畴分辨率，这些代数变体具有穿孔球对称幂的部分包裹的深谷范畴。这可能有助于找到Ozsvath-Szabo绝对结不变量的新结构。第二部分是将Hirzebruch-Riemann-Roch公式推广到非仿射簇和堆上的矩阵分解范畴。PI还想使用矩阵分解的范畴来找到g等变Gromov-Witten理论的Landau-Ginzburg对应物。项目的第三部分是在节点立方上实现非交换阶的结合型Yang-Baxter方程的三角解。第四部分是关于稳定超曲线的几何问题。研究了在稳定超曲线紧化边界附近超曲线模的Berezinian的Mumford同构类似的极点，并研究了在超曲线模空间上积分所引起的一些问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.