Moduli Spaces in Algebraic Geometry

代数几何中的模空间

• 批准号: RGPIN-2015-05631
• 负责人: Satriano, Matthew
• 金额: $ 1.6万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708661
• 关键词: Moduli; Spaces; Algebraic; Geometry

Arithmetic geometry is an important branch of mathematics that lies at the intersection of algebra, geometry, and number theory. The applications of the field are numerous, ranging from coding theory to cryptography to string theory. My research program is centered around several long-standing conjectures. The first of these is the Batyrev-Manin conjecture. It predicts in a precise way "how many" rational solutions there are to certain systems of polynomial equations. Although the conjecture is known in only a handful of cases, this far reaching statement greatly guides our thinking as to how rational solutions should behave in a wide level of generality. My program aims to extend the cases where this conjecture is known. A second conjecture concerns Hilbert schemes. These objects play a fundamental role in algebraic geometry, as they are often used to show the existence of other nice spaces, called moduli spaces. A longstanding problem within the field is to give a "rule" (moduli interpretation) that governs which algebras are on the main component of the Hilbert scheme of points. One of the biggest anticipated impacts of my research program for the mathematical community is the introduction of new tools with which to study the Hilbert scheme of points. The two main objects featured in my research program are stacks and Galois closures of ring extensions, as introduced by myself and Manjul Bhargava. Stacks are generalizations of the orbifolds showing up in theoretical physics. They are spaces equipped with additional structure encoding symmetries. One aspect of my research is the use of stacks in answering questions about ordinary spaces (ones with no additional stacky structure). Galois closures of ring extensions are highly accessible objects, and so lend themselves to projects at all levels. Accordingly, my research program will serve Canada through the training of highly qualified personnel.

算术几何是数学的一个重要分支，它是代数、几何和数论的交叉学科。该领域的应用是众多的，从编码理论到密码学再到弦理论。我的研究计划围绕着几个长期存在的问题。 第一个是Batyrev-Manin猜想。它以精确的方式预测某些多项式方程组有“多少”有理解。虽然这个猜想只在少数情况下被人知道，但这个意义深远的陈述极大地指导了我们的思考，即理性解在广泛的一般性水平上应该如何表现。我的程序旨在扩展已知这个猜想的情况。 第二个猜想涉及希尔伯特方案。这些对象在代数几何中扮演着重要的角色，因为它们经常被用来证明其他好空间的存在，称为模空间。一个长期存在的问题领域内是给一个“规则”（模解释），管理哪些代数的主要组成部分的希尔伯特计划的点。我的研究计划对数学界的最大预期影响之一是引入了研究希尔伯特点方案的新工具。 我的研究计划中的两个主要对象是栈和环扩展的伽罗瓦闭包，由我和Manjul Bhargava介绍。堆栈是理论物理中出现的轨道折叠的概括。它们是具有额外结构编码对称性的空间。我研究的一个方面是使用堆栈来回答关于普通空间（没有额外堆栈结构的空间）的问题。环扩展的伽罗瓦闭包是高度可访问的对象，因此适用于所有级别的项目。因此，我的研究计划将通过培养高素质的人才为加拿大服务。