BACH Seminar

Abstract: In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics via the geodesic flow of the right-invariant energy metric on the group of volume-preserving diffeomorphisms of the flow domain. We describe several recent ramifications of this approach related to compressible fluids, optimal mass transport, as well as Newton's equations on diffeomorphism groups and smooth probability densities. It turns out that various important PDEs of hydrodynamical origin can be described in this geometric framework in a natural way. This is a joint work with G.Misiolek and K.Modin.

Abstract: A rational blowdown surgery initially introduced by R. Fintushel and R. Stern and later generalized by J. Park is one of the simple but powerful techniques in the study of 4-manifolds topology. Note that a rational blowdown surgery replaces a certain linear chain of embedded 2-spheres by a rational homology 4-ball. In particular, a rational homology ball is a key ingredient in the construction of exotic smooth, symplectic 4-manifolds with small Euler characteristic and complex surfaces of general type with pg = 0. It also plays an important role in Q-Gorenstein smoothings and symplectic fillings of the link of surface singularities. In this talk, first I'd like to briefly review how to obtain symplectic fillings/Milnor fibers via a rational blowdown surgery. And then I'll investigate a relation between rational blowdowns and symplectic fillings/Milnor fibers on the link of weighted homogeneous surface singularities. This is a joint work with Hakho Choi.