PDE and Applied Mathematics Seminar[ Edit ]
Moderator: Yehuda Pinchover
Let f: R^n \to R^n be a Sobolev map. Suppose that the k-minors of df are smooth. What can we say about the regularity of f? This question arises naturally in the context of Liouville's theorem, which states that every weakly conformal map is smooth. I will explain the connection of the minors question to the conformal regularity problem, and describe a regularity result for maps with regular minors. If time permits, I will discuss these questions in the context of mappings between Riemannian manifolds.
The prediction of interactions between nonlinear laser beams is a longstanding open problem. A traditional assumption is that these interactions are deterministic. We have shown, however, that in the nonlinear Schrodinger equation (NLS) model of laser propagation, beams lose their initial phase information in the presence of input noise. Thus, the interactions between beams become unpredictable as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model.
Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) that result from differential equations with random input. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.