Moduli Spaces in Algebraic Geometry

算术几何是数学的一个重要分支，它位于代数、几何和数论的交叉点。该领域的应用非常广泛，从编码理论到密码学再到弦理论。我的研究项目围绕着几个长期存在的猜想。第一个是Batyrev-Manin猜想。它以一种精确的方式预测某些多项式方程系统有“多少”个有理解。尽管这一猜想仅在少数情况下为人所知，但这一影响深远的陈述极大地指导了我们思考理性解决方案在广泛的普遍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。第二个猜想是关于希尔伯特格式的。这些对象在代数几何中扮演着基本的角色，因为它们经常被用来显示其他漂亮空间的存在，称为模空间。该领域长期存在的一个问题是给出一个“规则”（模解释）来管理哪些代数是Hilbert点格式的主要组成部分。我的研究计划对数学界最大的预期影响之一是引入了研究希尔伯特点格式的新工具。***我研究项目中的两个主要对象是我和Manjul Bhargava介绍的环扩展的堆栈和伽罗瓦闭包。叠层是理论物理中出现的轨道的概括。它们是配备了额外结构编码对称的空间。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，因此适合所有级别的项目。因此，我的研究项目将通过培养高素质人才为加拿大服务

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.**