I will discuss the task of proving bounds on average quantities in dissipative dynamical systems, including time averages in finite-dimensional systems and spatiotemporal averages in PDE systems. In the finite-dimensional case, I will describe computer-assisted methods where bounds are proven by constructing nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proven by constructing Lyapunov functions. Nonnegativity of polynomials is enforced by requiring them to be representable as sums of squares, a condition that can be checked using the convex optimization technique of semidefinite programming. Rigorous bounds are obtained by supplementing numerical computations with either interval arithmetic or symbolic computation. These methods are illustrated using the Lorenz equations, where they produce novel bounds on various average quantities. In the PDE case, I will describe bounds for fluid dynamical models proven using pencil-and-paper analysis. Differences from the finite-dimensional case will be discussed, as well as the possibility of improving results for PDEs using computer-assisted methods like those described for the finite-dimensional case. Speaker(s): David Goluskin (University of Michigan)