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Research Interests

  • Nonlinear partial differential equations
  • Shape and structure of energy landscapes
  • Dynamic metastability or slow coarsening
  • Stochastic perturbations and rare events

Brief Description

I am interested in problems of applied analysis, often motivated by physics. Tools in my work include analysis, partial differential equations, and probability theory. I am particularly interested in the shape of complex energy landscapes and capturing the qualitative and quantitative behavior of the corresponding gradient flows, with and without the presence of a random perturbation.

Work related to the shape and dynamics of PDE energy landscapes includes the study of optimal relaxation rates for gradient flows with respect to a nonconvex energy, properties of the Cahn-Hilliard energy landscape for large systems and mean value close to -1, and dynamic metastability of gradient flows.

I have also studied the action minimization problem associated to the Allen Cahn partial differential equation. This is a time- and space-dependent problem in the Calculus of Variations. Here the numerical study of E, Ren, and Vanden-Eijnden was fundamental in discovering intriguing phenomena which were later explored in a series of papers together with Kohn, Otto, Tonegawa, Vanden-Eijnden; see also the extension by Mugnai and Röger. A second aspect of this research area concerns the interplay between small noise and large system size.