Foundations of Moduli Theory

本计画研究代数几何及相关领域的模空间。代数几何研究由多项式方程定义的空间的几何。除了是数学中最古老的学科之一，代数几何也是现代数学研究的前沿。它的抽象基础在现代数论、代数拓扑、表示论和组合学的研究中至关重要。同时，它在应用数学中提供了强大的工具，应用包括密码学，计算理论，凸优化，计算机图形学，统计学和机器学习。该项目旨在研究代数几何中最基本的问题之一，即代数簇的分类。模空间本身就是一个代数簇，它的点与被分类的代数簇一一对应。模空间为我们提供了关于被分类的几何对象的丰富信息，并且在许多其他数学领域（包括纯数学和应用数学）中有着深入的应用。研究目标是双重的：开发抽象的基础工具，模理论，然后应用这些工具来研究特定的模空间。本计画也将训练研究生进行模理论的研究，研究者发展出一种新的方法来建构具有正维自同构群的物体的投射模空间。虽然稳定曲线的模空间用有限自同构群对对象进行分类，但还有许多其他感兴趣的模空间不具有此特征。例子包括向量丛或层的模、Bridgeland半稳定复形的模和K-半稳定变种的模。最近的发展提供了一个代数栈的必要和充分条件，允许一个好的模空间的特征为0。这一结果已被用来构造Bridgeland半稳定对象和K-半稳定Fano簇的新的投射模空间。这种方法依赖于代数堆栈的局部结构定理，目前，仅限于特征0。该项目旨在将这些结果扩展到积极和混合的特征。同时，该项目旨在将最新进展应用于研究特定的模数空间，例如曲线模数空间的对数正则模型的模数描述。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

This project investigates moduli spaces in algebraic geometry and allied fields. Algebraic geometry studies the geometry of spaces defined by polynomial equations. In addition to being one of the most ancient subjects in mathematics, algebraic geometry is also at the forefront of research in modern mathematics. Its abstract foundations are vital in the study of modern number theory, algebraic topology, representation theory, and combinatorics. At the same time, it provides powerful tools in applied mathematics, with applications including cryptography, theory of computation, convex optimization, computer graphics, statistics, and machine learning. This project aims to study one of the most fundamental questions in algebraic geometry, namely classifying algebraic varieties. A moduli space is itself an algebraic variety whose points are in one-to-one correspondence with the algebraic varieties that are being classified. Moduli spaces provide us with rich information about the geometric objects being classified and moreover have deep applications in numerous other fields of mathematics, both pure and applied. The research objectives are twofold: to develop abstract foundational tools in moduli theory and then apply these tools to study specific moduli spaces. This project will also involve the training of graduate students in moduli theory research.The investigator has developed a new approach to construct projective moduli spaces of objects with positive dimensional automorphism groups. While the moduli space of stable curves classifies objects with finite automorphism groups, there are many other moduli spaces of interest that do not share this feature. Examples include the moduli of vector bundles or sheaves, the moduli of Bridgeland semistable complexes, and the moduli of K-semistable varieties. Recent developments have provided necessary and sufficient conditions for an algebraic stack to admit a good moduli space in characteristic 0. This result has already been applied to construct new projective moduli spaces of Bridgeland semistable objects and K-semistable Fano varieties. This approach rests on local structure theorems for algebraic stacks which, at the moment, are limited to characteristic 0. This project aims to extend these results to positive and mixed characteristics. At the same time, the project aims to apply recent advances to study specific moduli spaces of varieties such as modular descriptions of log canonical models of the moduli space of curves.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.