The analysis and geometry of isometric embedding
On Weyl Asymptotics
The renormalized volume on 4-dimensional CCE manifolds
徐国义：In 1950’s, Nash-Kuiper built up the C1 isometric embedding for any surface into R3, this can be viewed as analysis side of isometric embedding. On the other hand, there is obstruction for the existence of C2 isometric embedding of surface into R3 known since Hilbert, which reflects the geometry flavor of isometric embedding. What’s happening from C2 to C2 (from analysis to geometry)? The talk will be accessible to general audience with basic knowledge of analysis and geometry.
王作勤：Weyl law, first discovered by H. Weyl in 1911 for the Dirichlet-Laplace eigenvalues of bounded regions and then extended/strengthened by many mathematicians to various general settings, relates the asymptotic behavior of eigenvalues of certain operators with the background geometric/analytic/dynamic behavior. In this talk I will briefly describe these connections and discuss some recent work.
来米加：The renormalized volume is a very important global invariant for conformally compact Einstein (CCE) manifolds. In dimension 4, it is the integral of sigma_2 of the Schouten tensor, which appears in the Gauss-Bonnet-Chern formula. Based on Gursky’s work on the Weyl functional and the de Rham cohomology on closed 4-manifolds and Chang-Gursky-Yang’s conformal 4-sphere theorem, one can deduce interesting topological consequences for 4-dim CCE manifolds under assumptions on the renormalized volume. I will survey results in this direction and discuss some recent thoughts.