Derived Categories, Noncommutative Orders, and Other Topics

派生范畴、非交换顺序和其他主题

基本信息

批准号：
2001224
负责人：
Alexander Polishchuk
金额：
$ 23.9万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001224&HistoricalAwards=false
关键词：
Derived Categories Noncommutative Orders Other

项目摘要

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.

拟议的研究是在代数几何学领域，与字符串理论有一些联系。代数几何形状是研究由多项式方程和相关数学结构定义的数学对象的数学分支。从经典上讲，一个几何对象（称为代数品种）在它们上形成换向环（即可以添加并乘以乘以函数）的代数函数集合。现代研究涉及与代数品种相关的更复杂的代数结构，例如相干滑轮类别（类别的概念是关联环的概括）。该项目的一部分是建立某些同源镜对称性猜想的案例，该猜想确定了在两个看似无关的上下文中几何出现的类别。该项目的另一部分旨在为字符串理论中超级黎曼表面（通常表面的概括）的某些方面赋予严格的数学基础。该项目为本科和研究生提供了研究培训机会。更具体地说，该项目的第一部分是在刺穿球体的对称能力的同源镜子对称性上。目的是确定在某些代数品种上具有部分包裹的福卡亚类别的对称球体的对称能力的衍生类别类别的分类分辨率。这可能有助于找到Ozsvath-Szabo的分类结的新结构。第二部分是确定将Hirzebruch-Riemann-Roch公式概括为非植入品种和堆栈的矩阵因素化类别的类别。 PI还希望使用矩阵因法化类别来找到G-均等式Gromov-Witten理论的Landau-Ginzburg对应物。该项目的第三部分是实现阳性立方体的非交通订单方面的联想杨巴克斯特方程的三角解决方案。第四部分与稳定超级弯曲的几何形状有关。 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 标准。