Foundations of Moduli Theory

模理论的基础

• 批准号: 2100088
• 负责人: Jarod Alper
• 金额: $ 29.6万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100088&HistoricalAwards=false
• 关键词: Foundations; Moduli; Theory

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.

该项目研究代数几何和相关领域中的模空间。代数几何研究由多项式方程定义的空间几何。代数几何除了是数学中最古老的学科之一之外，也处于现代数学研究的前沿。它的抽象基础对于现代数论、代数拓扑、表示论和组合学的研究至关重要。同时，它提供了强大的应用数学工具，应用范围包括密码学、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。模空间本身就是一种代数簇，其点与正在分类的代数簇一一对应。模空间为我们提供了有关被分类的几何对象的丰富信息，并且在许多其他数学领域（无论是纯数学还是应用数学）都有深入的应用。研究目标有两个：开发模理论中的抽象基础工具，然后应用这些工具来研究特定的模空间。该项目还将涉及对模理论研究研究生的培训。研究者开发了一种新方法来构造具有正维自同构群的对象的射影模空间。虽然稳定曲线的模空间对具有有限自同构群的对象进行分类，但还有许多其他感兴趣的模空间不具有此特征。示例包括矢量束或滑轮的模数、Bridgeland 半稳定复合体的模数以及 K-半稳定簇的模数。最近的发展为代数栈在特征 0 上容纳良好的模空间提供了充分必要条件。该结果已应用于构造 Bridgeland 半稳定对象和 K-半稳定 Fano 簇的新射影模空间。这种方法依赖于代数栈的局部结构定理，目前该定理仅限于特征 0。该项目旨在将这些结果扩展到正特征和混合特征。同时，该项目旨在应用最新进展来研究变体的特定模空间，例如曲线模空间的对数正则模型的模块化描述。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。