Derived Categories, Noncommutative Orders, and Other Topics

拟议的研究是在代数几何领域与弦理论的一些连接。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，人们将这样的几何对象（称为代数簇）与它们上的代数函数的集合相关联，该集合形成交换环（即，函数可以相加和相乘）。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下，确定类别出现在两个看似无关的情况下的几何。该项目的另一部分旨在为弦理论中使用超级黎曼曲面（通常曲面的推广）的某些方面提供严格的数学基础。本计画提供研究训练机会给大学部及研究所学生，第一部份是关于穿孔球对称幂的同调镜像对称性。我们的目标是确定某些代数簇的部分包裹福谷类别的对称权力的穿孔领域的派生类别的相干层的类别决议。这可能有助于Ozsvath-Szabo范畴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栈上的矩阵分解范畴。PI也希望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。第三部分是利用节点三次体上的非对易阶实现结合型Yang-Baxter方程的三角解。第四部分是关于稳定超曲线的几何。PI提出了理解Mumford同构的类似物的极点的超曲线的模Berezinian的紧化的边界附近的稳定超曲线和研究在超曲线的模空间的集成中出现的一些问题。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.