Ioana Suvaina
Vanderbilt University

Einstein metrics and exotic smooth structures on 4-manifolds

The existence or non-existence of Einstein metrics on a topological 4-manifold is strongly related to the differential structure considered. We analyze this dependency for manifolds with small topology. We also consider 4-manifolds with the canonical smooth structure, and for a large class of manifolds we prove non-existence theorems of group invariant metrics. The main techniques used come from Seiberg-Witten theory, the geometry of complex surfaces and symplectic topology.

Hans Knüpfer
Courant Institute, NYU

Well-posedness and Lubrication Approximation for Spreading Droplets

Consider a 2-d droplet spreading on a plate. Our main interest is the moving triple point where air, liquid and solid meet. A simple model for the evolution of the liquid is given by the Darcy flow. In the regime of lubrication approximation, i.e. when the liquid film is very thin, the evolution can be described by a class of reduced 1--d models, the so called thin--film equations. We present well-posedness results for both the Darcy Flow and for certain thin-film equations. One difficulty is that the regularity at the moving triple point depends strongly on the imposed slip condition at the liquid-solid interface. In the regime of thin films, we show that solutions of the Darcy flow converge to solutions of the corresponding thin-film equation.

Juhi Jang
Courant Institute, NYU

Vacuum in Gas and Fluid dynamics

The study of vacuum is important in understanding the motion of a gaseous star or shallow water. The mathematical difficulty is the degeneracy caused by vacuum. In this talk, I will review some interesting problems of vacuum states arising in gas and fluid dynamics, and present the current status in the understanding of compressible Euler flows near vacuum. My contribution to the subject is joint with Nader Masmoudi.