I will present some recent work on the heat equation and the biharmonic heat equation on complete Riemannian manifolds, including: uniqueness criteria for the heat equation and the biharmonic heat equation, estimates of the biharmonic heat kernel, and a uniform L-infinite estimate for solutions of the biharmonic heat equation with bounded initial data.

We develop a dynamic tractable model where an investor derives realization utility as in Barberis and Xiong (2012) and Ingersoll and Jin (2013), but importantly can dynamically rebalance her portfolio between a risky asset and a risk-free asset. We show that the option of investing in the risk-free asset is quite valuable, even though the investor only derives utility from realized gains and losses of trading the risky asset. We also find that the investor may realize losses after a slight rally following a crash. This work is jointly with Cong Qin and Neng Wang

本报告主要讨论了空间周期介质中，双稳型方程周期行波解的波速对方向的依赖性。对于一维空间中的Fisher-KPP方程，由最小波速的变分表达式可知左右两个相反方向上的最小波速一定是相同的。但对于双稳方程，我们发现两个相反方向上的（唯一）波速可以相差很大。这个结论也被推广到高维情形和多稳定情形。在高维空间中，我们证明了在任意有限多个方向上，行波解波速可以相差很大，并由此指出：渐近传播速度可以严格小于行波解波速。在多稳态情形，我们发现左右两个方向上递归型行波解（propagating terrace）的形状、中间平衡态的个数、传播速度等都可以不同。报告是基于与Thomas Giletti 合作的工作。

We introduce a class of functional analogs of the symmetric difference metric on the space of coercive convex functions on R^n with full-dimensional domain. We show that convergence with respect to these metrics is equivalent to epi-convergence. Furthermore, we give a full classification of all isometries with respect to some of the new metrics. Moreover, we introduce two new functional analogs of the Hausdorff metric on the spaces of coercive convex functions and super-coercive convex functions, respectively, and prove equivalence to epiconvergence.

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.