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上。该领域的应用非常广泛，从编码理论到密码学再到字符串理论，应有尽有。我的研究计划围绕着几个长期存在的猜想展开。 第一个是拜特列夫-马宁猜想。它以一种精确的方式预测某些多项式方程组有多少有理解。尽管这一猜想只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的思考，即理性的解决方案应该如何在广泛的一般性水平上表现出来。我的程序旨在扩展这一猜想已知的情况。 第二个猜想与希尔伯特方案有关。这些对象在代数几何中起着基本的作用，因为它们经常被用来证明存在其他好的空间，称为模空间。该领域的一个长期存在的问题是给出一个“规则”(模解释)，该规则规定了哪些代数位于希尔伯特点列的主分量上。我的数学界研究计划最大的预期影响之一是引入了新的工具来研究点的希尔伯特方案。 我的研究计划中的两个主要对象是由我和Manjul Bhargava介绍的环扩张的堆栈和Galois闭包。堆栈是在理论物理中出现的奥比诺德的推广。它们是具有编码对称性的附加结构的空间。我的研究的一个方面是在回答关于普通空间(没有额外的堆叠结构的空间)的问题时使用堆栈。环扩张的Galois闭包是高度可访问的对象，因此适合于所有级别的项目。因此，我的研究计划将通过培训高素质的人员来服务于加拿大。

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.