Extremal problems in geometry

极值问题是要求在一定集合上优化给定数量的任何问题。这些问题在数学中无处不在。最著名的例子之一是等长问题，问给定周长的平面图形哪个面积最大，答案是一个圆盘。另一个经典的例子是球体填充问题，它问的是：d维球的哪种构型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时间，但它的解决方案只在维度1、2、3、8和24中已知。除了具有内在的趣味性外，极端问题对于确定特殊对象的存在也具有不可思议的强大作用。这种现象的例子包括fejsamir和Riesz对Riemann映射定理的证明，以及Bers对Thurston对曲面同胚映射类分类的证明，等等。我的研究计划的长期目标是研究和解决几何中的各种极值问题。虽然我们将主要关注双曲曲面，但我们也将考虑有限规则图和平面环面。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。对于这些物体中的任何一个，我们都可以将两个谱联系起来：长度谱（所有封闭测地线的长度集合）和拉普拉斯特征谱。这些光谱的研究产生了许多极端问题，例如：每个光谱中的第一个正入口及其多重度可以有多大？即使长度谱或特征谱中的任何一个决定了另一个，也没有明确的机制来做到这一点。相反，两个光谱之间的关系是由一个包含辅助测试函数的跟踪公式编码的。结果表明，人们可以通过寻找合适的测试函数来证明任意谱上的项上的普遍不等式。这就是2017年在8维和24维中解决球体填充问题的方法。在最近与索邦大学的合作者Bram Petri的合作中，我们将这种方法应用于双曲线表面。该项目的目标之一是利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界之外，我们还将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实时计算这些不变量并导航模空间，从而将问题用于数值优化。

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.