CAREER: Moduli Spaces and Derived Categories

数学和数学物理中的许多现象都可以用多项式方程组来描述。这样的方程组的解的集合可以具有非常复杂和有趣的形状。通常，方程涉及被视为“参数”的数字。当一个人改变这些参数时，确定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。本项目将介绍一种研究模问题的新方法，以及这种新方法在高能理论物理启发下的几个开放问题中的应用。PI将通过与本科生的研究项目，研究生的指导和针对K-12学生的讲座，致力于培养学生在这一研究领域。PI的目标之一是与涉及机器学习的行业建立联系。具体而言，首席研究员将使用代数堆栈理论和派生代数几何的最新发展，以制定一个通用的方法来解决代数几何中的模数问题。PI最近的工作扩展的方法和结果的几何不变理论的任意模的问题，导致了一个相对完整的框架，用于构建模空间和打破大模的问题成件，这是更容易研究。这个项目调查一些扩展的基础方面的这项工作，以及应用到几个问题的枚举几何。该项目还提出了应用程序，以研究派生类别的连贯sheaves.This奖项反映了NSF的法定使命，并已被认为是值得的支持，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.

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.