The ubiquity of optimal spectral gaps

ongoing

Spectral gap is a fundamental concept in mathematics, physics, and computer science as it governs the exponential rate at which a process converges towards its stationary state. It informs the spectral lines of hydrogen, how we shuffle cards, the behavior of semiconductors, and web search algorithms. Moreover, some of the most prominent issues of contemporary mathematics, including the Ramanujan-Petersson conjecture and the Yang-Mills mass gap, revolve around spectral gap. This proposal seeks to investigate the nature of the spectral gap for hyperbolic surfaces and unitary representations of fundamental groups of surfaces. In the former case, the spectral gap occurs in the spectrum of the Laplace-Beltrami operator on the surface, and in the latter, it occurs in the spectrum of a Hecke operator attached to the representation. The two main motifs of the proposal are ubiquity and optimality. Is the spectral gap ubiquitous? Does it exist for random surfaces and random representations? Is it easy to construct surfaces with a large spectral gap? In what cases can one prove that the spectral gap is close to optimal? The sharpest and most ambitious questions discussed in this proposal combine these two aspects and ask whether objects with (almost) optimal spectral gap appear with high frequency. My main technical tool is the development of new formulas for integration over representation varieties of fundamental groups of surfaces. These integral formulas are of high independent interest. For example, I propose to establish estimates that extend important results in Voiculescu's Free Probability Theory from the context of free groups, to fundamental groups of closed compact surfaces, and beyond. The proposal is extremely timely, as it builds on two separate breakthroughs that I have achieved in 2019. I am uniquely placed to tackle the questions of the proposal due to my broad background in geometry, analysis, and representation theory.