und via zoom (moodle)

Abstract:

From the simple harmonic oscillator to the Standard Model of particle physics, many quantum systems exhibit a finite gap between the ground state energy and that of the first excited state. This spectral gap can be traced back to the Bakry-Émery Laplacian-the natural self-adjoint, second-order differential operator on weighted Riemannian manifolds. The existence of such a gap reflects deep non-perturbative features of a quantum theory, including vacuum stability, exponential suppression of forces at large distances, and the absence of arbitrarily light particle excitations. It also plays a central role in the unsolved Millennium Prize problem: Yang-Mills & the Mass Gap.

In recent years, multiple authors (Moncrief, Marini, Maitra 2019; Mondal 2023) have pursued a geometric approach to spectral gap estimates by establishing lower bounds on the Bakry-Émery Ricci curvature. However, many physically significant systems-including Yang-Mills theory-involve non-smooth geometries arising from quotienting by symmetry group actions. In this talk, I explore how synthetic geometric techniques can be used to study the spectral gap in such non-smooth, symmetry-reduced systems. I begin with the classical Lichnerowicz bound and the harmonic oscillator as motivating examples, then explain the procedure of symmetry reduction in Hamiltonian systems, and conclude by reviewing synthetic results on spectral gap estimates and quotient spaces.