The semiclassical Schrodinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems.

With the development of financial markets, more and more complicated and advanced derivatives are designed to fit the market’s interests. There exotics products are often linked with multi-assets and under untraditional risk, e.g. correlation, credit-default-risk, etc.

As shown by Aronson in the 1960's, the fundamental solution of parabolic partial differential operators in divergence form can be bounded from above and below by the Gaussian heat kernel, i.e., the fundamental solution of the classical heat equation.

We will investigate various notions of hyperbolicity in Riemannian and complex geometries, with a focus on some conjectures (due to Hopf, Singer, etc.) on Euler characteristic of compact aspherical manifolds.

Based on Van Kan’s projection scheme, some fully decoupled schemes for incompressible fluid/thin-walled structure interaction were proposed. They show better accuracy than Fernandez’s schemes in 2D linear test case. 3D non-linear test case has not finished yet.

In this talk, I introduce a sharp gradient estimate for linear heat equation on complete non-compact Riemannian manifolds which can be considered as an extension of Hamilton’s gradient estimate. Moreover, I also give Liouville results for some nonlinear equations involving p-Laplacian.