We say a region of space is "cloaked" with respect to electromagnetic measurements if its contents -- and even the existence of the cloak -- are inaccessible to such measurements. One recent proposal for such cloaking takes advantage of the coordinate-invariance of Maxwell's equations. As usually presented, this scheme uses a singular change of variables. That makes the mathematical analysis subtle, and the practical implementation difficult. This talk examines the correctness and robustness of the change-of-variable-based scheme, for scalar waves modelled by Helmholtz's equation, drawing on joint work with Onofrei, Shen, Vogelius, and Weinstein. The central idea is to use a less-singular change of variables. The quality of the resulting "approximate cloak" can be assessed by studying the detectability of a small inclusion in an otherwise uniform medium. We show that a small inclusion can be made nearly undetectable (regardless of its contents) by surrounding it with a suitable lossy layer.

Consider extension operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. While such extensions are a fundamental tool in Sobolev space theory, they are essential in finite element analysis. In fact, for the latter, one needs extensions with additional polynomial preservation properties. The talk is devoted to recent developments in the construction of such extension operators.

We consider impulse control problems motivated from portfolio
optimization with sub-additive transaction cost. We show that the
optimal strategy exists and the number of its jumps is integrable. The
value function is characterized by a new type of Quasi-variational
inequalities. It is a joint work with Jin Ma, Jing Xu, and Jianfeng
Zhang.