My research interests lie in infinite-dimensional dynamical systems and nonlinear partial differential equations. I have been working on both the abstract theory of infinite-dimensional dynamical systems and also the concrete dynamics of particular nonlinear parabolic and hyperbolic PDEs. For instance, we established the existence of global in time bubble-like solutions to a nonlinear parabolic PDE and the existence, for long positive and negative time, of multi-peaked solutions to the Cahn-Hilliard equation. We also have a large project that, when completed, will include a large invariant manifold of solutions to the nonlinear Klein-Gordon equation. Another project I completed earlier was on patterned solutions in a diffusive Lengyel-Epstein system of CIMA chemical reactions. There we used bifurcation theory and rigorously proved the existence of spatially non-homogeneous periodic solutions and steady state solutions. Our abstract theorems concern the existence of center manifolds assuming one has
an approximately invariant manifold either with boundary or with a nontrivial center bundle for either maps or semi-flows in an infinite-dimensional space.
Since I joined MSU in 2009, I have taught a large variety of courses including College Algebra, Calculus IV in ODE, Calculus III in Multivariable Calculus, and Calculus I.