Moduli of A-Infinity Structures and Related Topics

A-无穷大结构的模及相关主题

基本信息

批准号：
1700642
负责人：
Alexander Polishchuk
金额：
$ 17万
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：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700642&HistoricalAwards=false
关键词：
Moduli Infinity Structures Related Topics

项目摘要

The project 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). The PI studies much more sophisticated algebraic structures associated with algebraic varieties, such as A-infinity algebras and derived categories of sheaves. In an A-infinity algebra, in addition to taking product of two elements, one has higher operations of multiplying together n-tuples of elements. These structures appeared independently in a very different way in works of Fukaya. The research is partly motivated by the desire to match different constructions of A-infinity algebras. The PI also intends to apply A-infinity structures to solve some problems in algebraic geometry.The project will focus mostly on the following two topics: moduli spaces of A-infinity structures and moduli spaces related to algebraic curves. In the first part of the project the goal is to construct moduli spaces (i.e., parameter spaces) of A-infinity structures related to various moduli spaces of geometric objects (sometimes involving non-commutative geometry). In particular, the PI plans to construct moduli spaces of A-infinity algebras related to algebraic surfaces, double covers of non-commutative projective lines, noncommutative orders over curves, as well as moduli spaces of A-infinity modules related to birational transformations. The second part of the project is devoted to studying toric GIT picture for moduli spaces of curves with nonspecial divisors, as well as moduli spaces of curves with vector bundles on them and their non-commutative analogs. Also, the PI plans to study natural vector fields on the moduli spaces of curves with nonspecial divisors and use them to produce explicit rational functions on the moduli spaces of curves with no marked points.

该项目属于代数几何领域，与弦理论有一些联系。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。传统上，人们将这些几何对象（称为代数簇）与它们上的代数函数集联系起来，这些函数形成交换环（即函数可以相加和相乘）。 PI 研究与代数簇相关的更复杂的代数结构，例如 A-无穷代数和滑轮的派生类别。在 A 无穷代数中，除了两个元素的乘积之外，还有将 n 元组元素相乘的高级运算。这些结构在深谷的作品中以一种截然不同的方式独立出现。这项研究的部分动机是希望匹配 A-无穷代数的不同构造。 PI还打算应用A-无穷大结构来解决代数几何中的一些问题。该项目将主要关注以下两个主题：A-无穷大结构的模空间和与代数曲线相关的模空间。在该项目的第一部分中，目标是构造与几何对象的各种模空间（有时涉及非交换几何）相关的 A-无穷大结构的模空间（即参数空间）。特别是，PI计划构造与代数曲面相关的A-无穷代数的模空间、非交换射影线的双重覆盖、曲线上的非交换阶，以及与双有理变换相关的A-无穷模的模空间。该项目的第二部分致力于研究具有非特殊除数的曲线模空间的环面 GIT 图，以及具有向量束的曲线模空间及其非交换类似物。此外，PI 计划研究具有非特殊约数的曲线模空间上的自然向量场，并使用它们在没有标记点的曲线模空间上生成显式有理函数。