This talk concerns a PDE system that models tumor growth. We show that a novel free boundary problem arises via the incompressible limit of this model. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. This is joint work with Inwon Kim.

We survey some new and old uniqueness results for Hessian equations such as special Lagrangian equations, Monge-Ampere equations, and symmetric Hessian equations. In particular, a unified approach to quadratic asymptote of solutions over exterior domains--based on an "exterior" Evans-Krylov, corresponding to Allard-Almgren's uniqueness of tangent cones in minimal surface situation--will be presented (joint with D.S. Li and Zh.S. Li). Special Lagrangian and Monge-Ampere equations are the potential equations for minimal and maximal "gradient" graphs in Euclid and pseudo-Euclid spaces respectively.

Note: This is a joint seminar with differential geometry.

The longstanding view in chemistry and biology is that high-affinity, tight-binding interactions are optimal for many essential functions, such as receptor-ligand interactions. Yet, an increasing number of biological systems are emerging that challenge this view, finding instead that low-affinity, rapidly unbinding dynamics can be essential for optimal function. These mechanisms have been poorly understood in the past due to the inability to directly observe such fleeting interactions and the lack of a theoretical framework to mechanistically understand how they work. In fact, it is only by tracking the motion of effector nanoprobes that afford detection of multiple such interactions simultaneously, coupled with inferences by stochastic modeling, Bayesian statistics, and bioimaging tools, that we recently begin to obtain definitive evidence substantiating the consequences of these interactions. A common theme has begun to emerge: rapidly diffusing third-party molecular anchors with weak, short-lived affinities play a major role for self organization of micron-scale living systems. My talk will demonstrate how these ideas can answer a longstanding question: how mucosal barriers selectively impede transport of pathogens and toxic particles.