Topic: spectral gaps

Ramon van Handel surveys recent progress on constructing graphs, manifolds and groups with optimal spectral gaps, which control how rapidly random walks explore a structure. The talk describes a polynomial method that proves optimality in new settings and connects these developments to longstanding questions about the prevalence of optimal gaps.

The Knabe method provides lower bounds on spectral gaps. This talk reviews the technique and applies it to random unitary circuits arranged in a one-dimensional brickwork pattern. For qubits, the resulting estimate is optimal up to a constant factor.

Singular configuration spaces require extensions of the usual operators and dynamics of quantum mechanics. Using metric-measure calculus, this work constructs states, observables and unitary evolution under suitable assumptions. A curvature-based technique bounds spectral gaps after symmetry reduction. Examples include a constrained oscillator, lattice Yang–Mills theory and fractional-dimensional spaces; the talk also considers prospective connections with the Yang–Mills mass-gap problem and gravity.