My research focuses on developing a global understanding of how complex systems change over time, and bridging the gap between what can be proven mathematically and what can be computed numerically.

Nonlinear differential equations are rarely explicitly solvable by hand. Instead of searching for arbitrary solutions, the dynamical systems viewpoint is to focus one's analysis on the qualitative behavior of invariant sets. For example, while a periodic orbit's geometry may be sensitive to perturbations, its topology (e.g. being homeomorphic to a circle) is much more robust. With abstract theorems one can describe in great detail the dynamics on and around generic invariant sets. However for a specific differential equation, verifying the hypotheses of such a theorem often requires hard quantitative analysis.

I am particularly interested in infinite dimensional dynamical systems and understanding their dynamics through a holistic study of a system's invariant sets. Computationally, this draws on a variety of numerical techniques from dynamical systems, partial differential equations, nonlinear optimization and algebraic topology. Analytically, this often involves proving theorems with explicitly verifiable hypotheses (e.g. rather than assuming "there exists some e>0", concretely quantifying how small e must be). The impetus for this is not bookkeeping for bookkeeping's sake. But rather to build a complete picture of a complex system from disparate pieces.

For example, standard numerical methods can solve an initial value problem for an ODE and provide local error bounds at each step. However a global error bound on the final solution requires the cumulative error be quantified. This quickly becomes a nontrivial problem in chaotic systems, where arbitrarily close initial conditions will inevitably diverge, and the difficulties compound in partial differential equations where the phase space is infinite dimensional.

To that end, validated numerics have been developed to keep track of all the sources of error inherent to numerical calculations. To bridge the gap between numerics and a computer assisted proof, a problem must be translated into a list of the conditions that the computer can check. Most famously used to solve the four color theorem, computer assisted proofs have been employed to great effect in dynamics, proving results such as the universality of the Feigenbaum constants and Smale's 14th problem on the nature of the Lorenz attractor.