My work studies both abstract theory and concrete examples of dynamical systems, with a strong emphasis on understanding chaotic (seemingly random or unordered) dynamical behavior. My background is in smooth ergodic theory, which provides a set of techniques and a conceptual framework (e.g., SRB measures) for understanding statistical properties of the time-asymptotic behavior of systems on smooth manifolds, the time-evolution of which is differentiable with respect to the initial condition. My work has typically focused on random and/or infinite dimensional driven dynamical systems.

Many of the time-evolving systems in our world evolve according to seemingly simple "microscopic" laws, e.g., Newton's 3rd law in the case of purely mechanical systems. It comes as some surprise, then, that these systems are capable of exhibiting extremely complicated and seemingly random behavior. Consider for example the gas surrounding you in the room you're in. At a microscopic level, the molecules comprising this gas interact with each other in a relatively simple (time-reversible) way, whereas at the macroscopic level the collective of gas molecules exhibits extremely complicated, time-irreversible behavior. Consider how smoke from incense distributes throughout a closed room-- have you ever seen it un-mix?

My research focuses on the statistical properties of these systems, i.e., how to understand the (possibly quite complicated or "chaotic") time-evolution of dynamics through the lens of probability theory (this goes by a different name-- ergodic theory).