Arithmetic and topology of moduli spaces

模空间的算术和拓扑

• 批准号: RGPIN-2019-05264
• 负责人: Groechenig, Michael
• 金额: $ 1.75万
• 依托单位: University of Toronto
• 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=715442
• 关键词: Arithmetic; topology; moduli; spaces

The mathematical landscape is divided into three big areas: algebra, analysis and geometry. Algebraic geometry is situated between algebra and geometry, and is concerned with the study of solutions of systems of polynomial equations in several variables. As the number of solutions is usually infinite, one can rarely compute all of them, and instead contents oneself with an understanding of the geometric shape formed by the ensemble of solutions. This is what we call an algebraic variety. It is the principal object of interest in algebraic geometry. Applications of this theory are wide-ranging: a special case, known as elliptic curves, plays a crucial role in the implementation of modern cryptography systems. Algebraic geometry also prominently appears in physics: string theory predicts that the four dimensions of space and time are supplemented by 6 extra dimensions which are curled up in the shape of a tiny algebraic variety (known as Calabi-Yau varieties). This project is devoted to the study of algebraic varieties known as moduli spaces. Rather than arising as the set of solutions to an explicit system of equations, each point of a moduli space represents a fixed type of mathematical objects, and describes its variations and deformations. Due to this geometric interpretation, the theory of moduli spaces is extremely rich and leads to beautiful applications in other areas of mathematics. We will study moduli spaces arising in algebraic geometry through tools provided by number theory. This allows us to confirm predictions originating in mathematical physics, in particular string theory. One of the main protagonists in my research is the moduli space of Higgs bundles. Together with my collaborators Dimitri Wyss and Paul Ziegler we proved a conjecture by Hausel and Thaddeus which relates the geometry of two such moduli spaces, related by Langlands duality. Their prediction was heavily influenced by mirror symmetry (a phenomenon observed in string theory), but our proof provides an entirely arithmetic approach. In a sequel to our proof of the Hausel-Thaddeus conjecture, we reverse the flow of ideas: our methods are used to give a new and elementary proof of the fundamental lemma (proven by Ngô in 2008). The latter is central to the Langlands programme, and hence to modern day understanding of number theory. In future work I will continue to explore this exciting connection between number theory, algebraic geometry and physics.

数学领域分为三大领域：代数、分析和几何。代数几何介于代数和几何之间，研究多变量多项式方程组解的问题。 由于解的数量通常是无限的，人们很少能计算出所有的解，而是满足于对由解的集合形成的几何形状的理解。这就是我们所说的代数族。它是代数几何中感兴趣的主要对象。 这一理论的应用范围很广：一种被称为椭圆曲线的特殊情况在现代密码系统的实现中起着至关重要的作用。代数几何也突出地出现在物理学中：弦理论预测，空间和时间的四个维度由6个额外维度补充，这些维度卷曲成一个微小的代数变种(称为Calabi-Yau变种)。 这个项目致力于研究被称为模空间的代数簇。模空间的每个点代表一种固定类型的数学对象，并描述其变化和变形，而不是作为显式方程组的解的集合产生的。由于这种几何解释，模空间的理论是极其丰富的，并导致在数学的其他领域的美丽应用。我们将通过数论提供的工具来研究代数几何中的模空间。这使我们能够证实起源于数学物理的预测，特别是弦理论。 我研究的主要主角之一是Higgs丛的模空间。与我的合作者Dimitri Wyss和Paul Ziegler一起，我们证明了Hausel和Thaddeus的一个猜想，该猜想与两个这样的模空间的几何有关，这两个模空间由朗兰兹对偶关系。他们的预测受到镜像对称性(弦理论中观察到的一种现象)的严重影响，但我们的证明提供了一个完全算术的方法。 在Hausel-Thaddeus猜想证明的续篇中，我们颠倒了思想的流动：我们的方法被用来给出基本引理的一个新的初等证明(由Ngôin在2008年证明)。后者是朗兰兹计划的核心，因此也是现代数论理解的核心。在未来的工作中，我将继续探索数论、代数几何和物理之间的这种令人兴奋的联系。