CAREER: Moduli Spaces and Derived Categories

数学和数学物理中的许多现象都是用多项式方程组来描述的。这样一个方程组的解的集合可以有一个非常复杂和有趣的形状。通常情况下，这些方程涉及被视为“参数”的数字。当改变这些参数时，确定一组解的几何性质如何变化的问题被称为“模问题”，模问题对代数和几何中的许多学科都很重要。本项目将引入一种研究模问题的新方法，并将这种新方法应用于受高能理论物理启发的几个开放问题。PI将通过与本科生合作的研究项目、对研究生的指导以及针对k-12学生的讲座，致力于培养这一研究领域的学生。PI的目标之一是与涉及机器学习的行业建立联系。具体来说，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代数几何中模问题的一般方法。PI最近在将几何不变理论的方法和结果推广到任意模问题上的工作，已经为构造模空间和将大模问题分解成更容易研究的小块提供了一个相对完整的框架。这个项目研究了这项工作的一些基础方面的扩展，以及在枚举几何中的几个问题的应用。该项目还提出了应用于研究相干束的派生类别。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。

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.