AI-Assisted Exploration of the Polygonal Faber-Krahn Inequality

Mentor: Dr. Levon Nurbekyan

Overview

This project focuses on the polygonal Faber-Krahn inequality, which conjectures that among all $n$-gons of fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue of the Laplacian. The conjecture was introduced by Polya and Szego, who proved it for $n=3,4$ using Steiner symmetrization. For $n \ge 5$, however, these techniques break down, and the conjecture remains open.

The goal of the project is to use AI-assisted methods to explore this problem both computationally and conceptually, with the aim of either identifying potential counterexamples or producing empirically supported conjectures and strategies that could inform future theoretical work.

Computational Approach

For a broad range of values of $n$, we will numerically search for candidate optimal $n$-gons using a combination of gradient-based optimization and reinforcement learning, and subsequently leverage evolutionary coding agents in the spirit of AlphaEvolve to refine and extend these optimization strategies. The resulting large-scale experiments will either provide numerical evidence supporting the conjecture or identify configurations that warrant closer scrutiny.

Analysis

In the absence of apparent counterexamples, we will use models with advanced reasoning capabilities, in the spirit of DeepThink, to analyze the numerical results at a higher level, seeking recurring patterns or structural features in the evolution of optimizing shapes that may suggest new geometric or analytical approaches to the problem.

Prerequisites

A solid background in linear algebra, multivariable calculus, and basic numerical methods.