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.

Correlators of a rational conformal field theory (RCFT) can be described as specific elements in spaces of conformal blocks.

Can machines generate music? Recent deep learning-driven systems have made breakthroughs on this topic. Some famous among them are Coconet, DeepBach, GLSR-VAE, C-RNN-GAN, etc. However the generated music is still not satisfying based on the feedbacks from different music communities.