CAREER: Moduli Spaces and Derived Categories

职业：模空间和派生范畴

基本信息

批准号：
1945478
负责人：
Daniel Halpern-Leistner
金额：
$ 40万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945478&HistoricalAwards=false
关键词：
CAREER Moduli Spaces Derived Categories

项目摘要

Many phenomena in mathematics and mathematical physics are described by systems of polynomial equations. The set of solutions of such a system of equations can have a very complicated and interesting shape. Typically the equations involve numbers which are regarded as “parameters.” The problem of determining how the geometric properties of the set of solutions change as one varies these parameters is called a “moduli problem,” and moduli problems are important to many subjects in algebra and geometry. This project will introduce a new approach to studying moduli problems, as well as applications of this new approach to several open problems inspired by high energy theoretical physics. The PI will work towards the training of students in this research field, through research projects with undergraduate students, mentoring of graduate students, and lectures aimed to k-12 students. One of the PI's goals is to build connections with industry where machine learning is involved. Specifically, the Principal Investigator will use recent developments in the theory of algebraic stacks and derived algebraic geometry to develop a general approach to moduli problems in algebraic geometry. The PI’s recent work on extending the methods and results of geometric invariant theory to arbitrary moduli problems has led to a relatively complete framework for constructing moduli spaces and for breaking large moduli problems into pieces which are easier to study. This project investigates some extensions of the foundational aspects of this work, as well as applications to several problems in enumerative geometry. The project also proposes applications to studying derived categories of coherent sheaves.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.

数学和数学物理中的许多现象都是通过多项式方程组来描述的，此类方程组的解集通常具有非常复杂且有趣的形状，这些方程涉及被视为“参数”的数字。确定一组解的几何性质如何随着这些参数的变化而变化，称为“模问题”，模问题对于代数和几何中的许多学科都很重要，该项目将引入一种研究模问题的新方法，如以及这个的应用受高能理论物理学启发，PI 将致力于通过本科生的研究项目、研究生的指导以及针对 k-12 学生的讲座来培训该研究领域的学生。 PI 的目标是与涉及机器学习的行业建立联系，具体而言，首席研究员将利用代数堆栈理论和派生代数几何的最新发展来开发解决代数几何中模问题的通用方法。 PI 最近将几何不变量理论的方法和结果扩展到任意模问题，从而形成了一个相对完整的框架，用于构造模空间并将大模问题分解为更容易研究的部分。该项目研究了基础模型的一些扩展。这项工作的各个方面，以及反映枚举几何中的几个问题的应用，该项目还提出了研究相干滑轮的派生类别的应用。该奖项是 NSF 的法定使命，并通过使用基金会的评估被认为值得支持。智力价值和更广泛的影响审查标准。