Extremal problems in geometry

几何中的极值问题

• 批准号: RGPIN-2022-03649
• 负责人: FortierBourque, Maxime
• 金额: $ 1.75万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759367
• 关键词: Extremal; problems; geometry

An extremal problem is any problem asking to optimize a given quantity over a certain set. These problems are ubiquitous in mathematics. One of the most famous examples is the isoperimetric problem asking which planar figure with a given perimeter has the largest area, the solution being a round disk. Another classical example is the sphere packing problem which asks: Which configurations of d-dimensional balls fill Euclidean space the most efficiently? Despite the fact that this problem has been around for a long time, its solution is only known in dimensions 1, 2, 3, 8, and 24. In addition to being intrinsically interesting, extremal problems can be incredibly powerful for establishing the existence of special objects. Examples of this phenomenon include Fejér and Riesz's proof of the Riemann mapping theorem as well as Bers's proof of Thurston's classification of mapping classes of surface homeomorphisms, among many others. The long-term goal of my research program is to study and solve a variety of extremal problems in geometry. While we will mostly focus on hyperbolic surfaces, we will also consider finite regular graphs and flat tori. In fact, our investigations will be largely guided by analogies between these objects. For any of these objects, one can associate two spectra: the length spectrum (the set of lengths of all closed geodesics) and the eigenspectrum of the Laplacian. Many extremal problems arise from the study of these spectra, such as: How large can the first positive entry and its multiplicity be in each spectrum? Even though either one of the length spectrum or the eigenspectrum determines the other, there is no explicit mechanism for doing so. Instead, the relationship between the two spectra is encoded by a trace formula, which involves an auxiliary test function. It turns out that one can prove universal inequalities on the entries in either spectrum by finding appropriate test functions. This is how the sphere packing problem was solved in dimension 8 and 24 in 2017. In recent work with collaborator Bram Petri from Sorbonne Université, we have adapted this method to hyperbolic surfaces. One of the objectives of this project is to extract the best bounds possible from it using numerical tools. In addition to proving new upper bounds for various geometric invariants, we will work on finding examples where these invariants are as large as possible. To help with this task, we will develop computer programs to calculate these invariants in real time and to navigate moduli space, lending the problem to numerical optimization.

极值问题是任何要求在特定集合上优化给定数量的问题。这些问题在数学中是普遍存在的。最著名的例子之一是等周问题，即给定周长的平面图形的面积最大，其解是一个圆盘。另一个经典的例子是球填充问题，它问：d维球的哪些配置最有效地填充欧几里得空间？尽管这个问题已经存在了很长一段时间，但它的解决方案仅在1，2，3，8和24维中已知。 除了本质上有趣之外，极值问题对于建立特殊对象的存在性也是非常强大的。这一现象的例子包括费耶尔和Riesz的证明黎曼映射定理以及Bers的证明瑟斯顿的分类映射类的表面同胚，等等。我的研究计划的长期目标是研究和解决各种几何极值问题。虽然我们将主要关注双曲曲面，但我们也将考虑有限正则图和平坦环面。事实上，我们的研究将在很大程度上以这些物体之间的类比为指导。对于任何一个这样的天体，我们都可以将两个谱联系起来：长度谱（所有闭测地线的长度集合）和拉普拉斯算子的本征谱。许多极端的问题出现在这些光谱的研究，如：多大的第一个积极的项目和它的多重性可以在每个频谱？ 即使长度谱或本征谱中的一个决定另一个，也没有明确的机制来这样做。相反，两个光谱之间的关系是由一个迹公式编码的，它涉及一个辅助测试函数。事实证明，人们可以通过找到适当的测试函数来证明任一谱中的条目上的泛不等式。这就是2017年在8维和24维中解决球体填充问题的方法。在最近的工作与合作者布拉姆Petri从索邦大学，我们已经适应了这种方法的双曲曲面。该项目的目标之一是使用数值工具从中提取最佳边界。除了证明各种几何不变量的新上限外，我们还将努力寻找这些不变量尽可能大的例子。为了帮助完成这项任务，我们将开发计算机程序来计算这些不变量在真实的时间和导航模空间，贷款问题的数值优化。