Advances in Moduli Spaces and Algebraic Stacks

研究者将研究代数几何及相关领域内的各种问题。代数几何是数学中最古老的学科之一，它的起源可以追溯到2400多年前希腊数学家对曲线的代数研究。大约60年前，随着格罗滕迪克开创性的工作，代数几何发生了翻天覆地的变化。这种转变带来了更严格和更广泛的基础，以及非常强大的技术，这些技术被用来解决许多长期存在的猜想，不仅在代数几何中，而且在其他数学领域，如数论、拓扑和统计学中。最近，在过去的几十年里，代数几何的范围得到了进一步的扩展，产生了一系列令人震惊的实际应用。代数几何现在为管理在线交易的密码系统、电影、游戏和虚拟现实中使用的计算机图形学、生物系统研究中的计算方法和药物治疗的有效性以及包括数据科学、机器学习和计算机视觉在内的许多其他领域提供了技术支柱。该项目旨在研究围绕模空间概念的广泛思想集合。在代数几何中，几何直觉和证明结果所需的技术工具之间存在很大的差距，这种差距可能不比模空间的研究更大。用于研究模空间的一个基本但高度技术性的工具是代数堆栈理论。研究者将进一步发展代数堆栈的基础，特别是通过发展一种内在方法来构造射影模空间，参数化可能具有正维自同构群的对象。研究者将继续研究这些基础结果的应用，以研究模空间的双空间几何，如由镜像对称激发的d等价猜想。最后，研究者将尝试应用复杂的代数几何技术来解决代数复杂性理论中的问题。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。

The investigator will research various problems within the field of algebraic geometry and related fields. Algebraic geometry is one the most ancient subjects in mathematics with its origins traced to the algebraic study of curves by Greek mathematicians more than 2400 years ago. Algebraic geometry drastically transformed around 60 years ago with the seminal work of Grothendieck. This transformation led to more rigorous and much broader foundations with immensely powerful techniques which were employed to resolve a number of long-lasting conjectures not only within algebraic geometry but also other mathematical fields such as number theory, topology and statistics. Most recently, the last few decades have seen the reach of algebraic geometry expand even further with an astounding array of practical applications. Algebraic geometry now provides the technical backbone to cryptosystems governing online transactions, to computer graphics used in movies, games and virtual reality, to the computational methods in the study of biological systems and the effectiveness of drug treatments, and to many other fields including data science, machine learning and computer vision. This project aims to study a wide collection of ideas encircling the concept of moduli spaces. In algebraic geometry, there is a large disparity between geometric intuition and the technical tools needed to prove results, and this disparity is perhaps no greater than in the study of moduli spaces. A fundamental yet highly technical tool used to study moduli spaces is the theory of algebraic stacks. The investigator will further the development of the foundations of algebraic stacks by in particular developing an intrinsic method to construct projective moduli spaces parameterizing objects that may have positive dimensional automorphism groups. The investigator will pursue applications of these foundational results to study the birational geometry of moduli spaces such as the D-equivalence conjecture motivated by mirror symmetry. Finally, the investigator will attempt to apply sophisticated algebro-geometric techniques to tackle questions in algebraic complexity theory.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.