The Center for Applied Mathematical Sciences is an organized research unit
based in the Department of Mathematics
The purpose of CAMS is to foster research and graduate education in
Mathematics in a broad sense and in an interdisciplinary mode. One goal
of the center's participants is to facilitate and encourage the development of
applicable mathematics and its utilization in problems in engineering and the
Winners of the CAMS Graduate Student Prize for Excellence in Research with a Substantial Mathematical Component.
More than 25 years ago, Kenig, Ruiz, and Sogge established uniform $L^p$ resolvent estimates for the Laplacian in the Euclidean space. Taking their remarkable estimate as a starting point, we shall describe more recent developments concerned with the problem of controlling the resolvent of elliptic self-adjoint operators in $L^p$ spaces in the context of a compact Riemannian manifold. Here some new interesting difficulties arise,...
The interest in and study of compact, co-dimension one minimizers has at least a century-old history: in 1904, J. J. Thomson proposed minimizing the electrostatic potential over sets of particles restricted to a sphere as part of his model of the atom. Modern physical examples of these assemblies occur in the realm of interacting nanoparticles. Many species of virus rely on the formation of a hollow sphere to enclose and deliver...
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data has been one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions...
We consider the problem of resonances for Schroedinger's equation and Helmholtz equation. These equation provide a model for an optimal design problem in which the goal is to create an optical structure that has a resonance that has low loss. In addition we will also discuss an asymptotic method for calculating low-loss resonances which is the forward problem in this inverse design problem.