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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613656
• 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.

算术几何形状是数学的重要分支，位于代数，几何学和数理论的交集。该领域的应用是众多的，从编码理论到密码学到字符串理论。我的研究计划集中在几个长期的猜想围绕。 其中的第一个是batyrev-manin猜想。它以一种精确的方式“有多少”理性解决方案对某些多项式方程式有多少理性解决方案。尽管猜想仅在几个案例中才知道，但这种遥不可及的陈述极大地指出了我们关于理性解决方案应如何以广泛的一般性的看法。我的计划旨在扩大已知猜想的情况。 第二个猜想涉及希尔伯特计划。这些对象在代数几何形状中起着基本作用，因为它们通常用于显示其他不错的空间的存在，称为模量空间。该领域内的一个长期存在的问题是给出一个“规则”（模量解释），该问题控制着哪个代数位于希尔伯特点方案的主要组成部分。我的研究计划对数学社区的最大预期影响之一是引入了研究希尔伯特（Hilbert）方案的新工具。 我和Manjul Bhargava介绍的我的研究计划中的两个主要对象是堆栈和GALOIS关闭。堆栈是在理论物理学中出现的Orbifolds的概括。它们是配备了其他编码对称性结构的空间。我的研究的一个方面是使用堆栈回答有关普通空间的问题（没有其他堆栈结构）。环形扩展的封闭是高度可访问的对象，因此在各个级别的项目中都提供了自己的限制。因此，我的研究计划将通过培训高素质的人员为加拿大服务。