In this talk, we will present couple of recent works about the convex floating bodies of equilibrium, asking whether there exists a non-ball convex body floating in the water in equilibrium position.

Let F be a family of sets in R^d. A set M〖⊂R〗^d is called F- convex if for any pair of distinct points x,y∈M, there is a set F∈F such that x,y∈F and

It is a survey talk on the Cheeger and Gromoll's well-known Soul Theorem, which states that any noncompact complete manifold M with nonnegative sectional curvature contains a compact totally geodesic submanifold Σ in M, called the soul of M, whose normal bundle is diffeomorphic to M. As a consequence, all the topology of M is concentrated in Σ, therefore is of finite type. Cheeger and Gromoll also conjectured that if M is of strictly positive curvature at one point, then the soul is also a point (thus M is diffeomorphic to the Euclidean space). The solution of this Soul Conjecture is one of G. Perelman's famous works, where some essential geometric structures from M to the soul were established. The soul theorem provides a guideline in understanding of manifolds with various curvatures, e.g., why the unsolved Milnor conjecture is reasonalble for nonnegative Ricci curvature. Many works have been invoked by Cheeger-Gromoll and Perelman's work.

In this talk, we will review the classical Gehring’s linked sphere problem. Gage’s solution to this problem will be sketched. We will also discuss Gromov’s point of view of this problem and his approach to the isoperimetric inequality in an infinite dimensional Banach space. Our spherical version of this linked sphere problem will also be mentioned. 腾讯会议：https://meeting.tencent.com/s/vjvTdiIhDfQv

In this talk, I will survey the classical results on blow-up analysis of 2d harmonic maps, including the bubble-tree construction. Then I will introduce some recent progress in this area.

This talk concerns the global existence of smooth sonic-supersonic potential flows in a two-dimensional expanding nozzle with the critical geometry at the inlet. The flow is governed by a quasilinear non-strictly hyperbolic equation with degeneracy at the inlet. An interesting phenomenon is that the existence of such sonic-supersonic flows depends on the height of the inlet. These results, together with other works, describe completely the geometry of the de Laval nozzles where there are smooth transonic flows of Meyer type whose sonic points are all exceptional.

Coding theory is a research field that spans across mathematics, computer science and engineering. It makes use of classical and modern algebraic techniques involving finite fields, group theory, polynomial algebra, etc, and it is concerned with metric structures of codes. Constructing codes from shorter ones and investigating their properties via those of shorter ones is an important and hot topic in coding theory. Blackmore and Norton (2001) introduced the notion of matrix product codes over finite fields, which is a generalization of many well-known constructions of codes, such as the (u|u + v)-construction, etc. In this talk, we will summarize some recent works on matrix product codes.

We study the dynamics of reaction-diffusion-advection models for single and two competing species in one-dimensional periodic habitats, where the individuals are subject to both diffusion and advection. We establish the monotone dependence of the principal eigenvalue on diffusion and drift rates. As applications, we first consider the persistence and spatial spreading of a single species and establish the critical threshold for the persistence as well as the monotone dependence of the minimal wave speed on the drift rate. We also consider two competing species model and study the local and global stability of semi-trivial steady states. Furthermore, the existence of evolutionarily singular strategies is established, which helps gain deeper insight into the evolution of dispersal in advective environments. This is a joint work with Shuang Liu.