Advances in Moduli Spaces and Algebraic Stacks

模空间和代数栈的进展

• 批准号: 1801976
• 负责人: Jarod Alper
• 金额: $ 15万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801976&HistoricalAwards=false
• 关键词: Advances; Moduli; Spaces; Algebraic; Stacks

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.

研究人员将研究代数几何和相关领域领域内的各种问题。 代数几何形状是数学中最古老的主题之一，其起源可追溯到2400年前希腊数学家对曲线的代数研究。 代数几何形状大约在60年前，通过Grothendieck的开创性作品进行了巨大变化。 这种转变导致了更加严格，更广泛的基础，其功能强大的技术不仅在代数几何形状内，而且还可以解决许多长期的猜想，还可以解决其他数学领域，例如数字理论，拓扑和统计。 最近，过去几十年来，随着令人震惊的实用应用程序，代数几何形状的影响力进一步扩大。 现在，代数几何形状为管理在线交易的密码系统提供了技术骨干，向电影，游戏和虚拟现实中使用的计算机图形技术，用于生物系统研究的计算方法以及药物治疗的有效性，以及包括数据科学，机器学习和计算机视觉的许多其他领域。该项目旨在研究包围模量空间概念的广泛想法。 在代数几何形状中，几何直觉与证明结果所需的技术工具之间存在很大的差异，并且这种差异也许不比模量空间的研究大。 用于研究模量空间的基本技术工具是代数堆栈的理论。 研究者将通过特别开发一种固有的方法来构建可能具有正数自动晶体的参数化对象，以进一步开发代数堆栈的基础。研究者将追求这些基本结果的应用，以研究模量空间的生育几何形状，例如由镜子对称性动机的D-等效性猜想。 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NSF的法定任务，并被认为是通过基金会的智力优点和更广泛的影响来通过评估来获得支持的。