In this talk, we will introduce some regularity results for Dirichlet problems with free boundary on Alexandrov spaces with curvature bounded from below. It contains the Lipschitz regularity of solutions and the finite perimeter property of their free boundary. This is based on a joint work with Chung-Kwong Chan, and Xi-Ping Zhu.

In this talk, we establish W^{2,p} estimates for elliptic equations on C^{1,\alpha} domains. The classical method, straightening the boundary, is not applicable since the domain is not C^{1,1} which is the standard assumption to derive W^{2,p} estimates. Both Vitali cover lemma (or C-Z decomposition) and Whitney cover lemma are used. An interesting property of harmonic functions is crucial to our result.

We consider a class of structured nonsmooth difference-of-convex minimization. We allow non-smoothness in both the convex and concave components in the objective function, with a finite max structure in the concave part. Our focus is on algorithms that compute a (weak or standard) directional-stationary point as advocated in a recent work of Pang et al. (Math Oper Res 42:95–118, 2017). Our linear convergence results are based on direct generalizations of the assumptions of error bounds and separation of isocost surfaces proposed in the seminal work of Luo and Tseng (Ann Oper Res 46–47:157–178, 1993), as well as one additional assumption of locally linear regularity regarding the intersection of certain stationary sets and dominance regions.

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.