Extremal problems in geometry

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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.