Ph.D. Mathematics. University of Texas at Austin. December 2009
Computational Topology in Dynamical Systems: Conley Index Theory, Set Oriented Numerics, computational aspects of Morse/Floer theory, and Applications.
A Posteriori Analysis and Numerics in Dynamical Systems: Parameterization of invariant manifolds with rigorous error bounds. Applications to computer assisted proof of connecting dynamics for diffeomorphisms, ordinary and partial differential equations.
KAM Theory: existence and numerical approximation of invariant manifolds in spite of small divisors.
The computer as the mathematician's "laboratory" for studying global dynamics of nonlinear systems. I'm especially interested in the computation and visualization of smooth invariant manifolds.
Reformulation of qualitative questions about nonlinear dynamical systems into quantitative functional equations, and numerical methods for studying these functional equations.
Rigorous numerical methods and computer assisted proof in analysis, especially constructive a-posteriori existence (or "shadowing") theorems for invariant manifolds, connecting dynamics, and chaotic motions.
Analytic continuation of local (un)stable manifolds with rigorous computer assisted error bounds.'' With Shane Kepley and Bill Kalies. SIAM Journal on Applied Dynamical Systems, Vol 17, No. 1, pp. 157--202 (2018)
Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications.'' With Maxime Murray. International Journal of Bifurcation and Chaos, Vol 27, No. 14 (2017).