Our research primarily deals with applications of spectral theory, with two main general themes. First, we study how estimates on the density of the spectrum of first-order differential operators may lead to explicit rates of convergence in the associated ergodic theorem. Separately, we are interested in how to actually compute (numerically, say) the spectrum of an operator. This leads to a very general question about how to compute “infinite” quantities starting from finite-dimensional approximations.
The importance of the spectrum is nicely expressed in John von Neumann’s proof of the Quasi-Ergodic Hypothesis:
The pith of the idea in Koopman’s method resides in the conception of the spectrum reflecting, in its structure, the properties of the dynamical system—more precisely, those properties of the system which are true “almost everywhere,” in the sense of Lebesgue sets.
John von Neumann, 1932