In 1957, Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2^n translations of its interior. The Hadwiger’s covering functional gamma_m(K) is the smallest positive number r such that K can be covered by m translations of rK. Due to Zong’s program, we study the Hadwiger’s covering functional for the simplex and the cross-polytope. In this talk, we will show the new upper bounds for the Hadwiger’s covering functional of the simplex and the cross-polytope, together with some other cases.

Let R be a discrete valuation ring with fraction field K. In this talk, we consider degenerations of a class of abelian varieties over K, namely the ones which admit good reduction over a tamely ramified finite field extension of K. They extend uniquely to “ket log abelian schemes” over R which is regarded as a log scheme with respect to the log structure associated to a chosen uniformizer of R. We will first give a brief introduction to log schemes and Kummer log etale (ket) topology. Then we define ket abelian schemes to be ket sheaves which are (ket) locally just abelian schemes. At last we state and show our degeneration result.

Discrete conformal structure is a discrete analogue of the smooth conformal structure on manifolds. There are different types of discrete conformal structures that have been extensively studied in the history, including the tangential circle packing, Thurston's circle packing, inversive distance circle packing and vertex scaling on surfaces, sphere packing and Thurston's sphere packing on 3-dimensional manifolds. In this talk, we will discuss some recent progresses on the rigidity of discrete conformal structures on two- and three-dimensional manifolds, including Glickenstein’s conjecture on the rigidity of discrete conformal structures on surfaces and Cooper-Rivin’s conjecture on the rigidity of sphere packings on three dimensional manifolds.

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