Mind the gap

Many useful properties of operators can be expressed in terms of their spectral gaps, or the difference in their two smallest eigenvalues. For instance, the spectral gap is relevant to bounding the runtime of an adiabatic optimization algorithm and mixing times of (sub-)stochastic processes, understanding the isoperimetric profile of spaces and rates of heat diffusion, and quantum phase transitions. In this talk, I will briefly introduce the audience to the utility of the spectral gap in various physical, mathematical, and computational situations and certain (often overlooked) equivalences between them. Time permitted, I will state the results of some known sub-optimal methods for bounding the spectral gap, our recent optimal approach, and inherent difficulties to determining rigorous bounds.