In this talk, two geometric variational problems, so-called partitioning problems for bounded domains will be discussed. Type-I is on area minimizing hypersurfaces with volume constraint and Type-II is on area minimizing ones with wetting area constraint. The stationary points for Type-I are free boundary CMC hypersurfaces while the ones for Type-II are minimal hypersurfaces with constant contact angle. When the bounded domain is a Euclidean ball, the global minimizers have been classified in 1970's. We will talk about the classification of the local minimizers for these problems. Moreover, we give Morse index estimate for such variational problems.

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in R^3, as an application we get the Brunn–Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in R^3.

题目：An Introduction to Stability of Traveling Wave Solutions in Free Interface Problems报告人：Claude-Michel Brauner (Tongji and Bordeaux (France) Univ.)地点：致远楼108室时间：2020-11-19, 12-03, 12-17（星期四）上午8：00—10：00In this round of talks, we intend to discuss a method especially suitable for the analysis of stability of traveling waves in free interface problems. For convenience, ...

We will talk about Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star-shaped hypersurfaces in Reissner-Nordstrom-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds in which can be realized as graphs over Reissner-Nordstrom-anti-deSitter manifold.

由一列球或矩形确定的上极限集是丢番图逼近中两类基本的集合, 一个源自于Dirichlet定理另一个源自于Minkowski定理. 由球确定的上极限集的度量理论已经非常丰富/完备, 然而由矩形确定的上极限集的研究却非常滞后, 甚至一些基本问题都尚未完全解决. 在此报告中, 通过引入“矩形的无处不在性/满测性”, 我们确定了由矩形生成的上极限集的Hausdorff理论的一般原理.

2013年，T. Bridgeland利用投射模的2-周期复形的 Ringel-Hall代数的局部化实现了量子包络代数的整体。之后，M. Gorsky分别用两种方法推广 T. Bridgeland的构造来定义所谓的半导出 Hall代数，一种是用 Frobenius范畴，其构造与 T. Bridgeland的没有本质区别，另一种要求比较强的条件并且其构造不是很直接。注意到他们的构造在2-周期复形的情形下都不能包含加权射影线上凝聚层范畴这类重要的遗传范畴，与卢明合作我们对任意遗传范畴的2-周期复形范畴定义了 modified Ringel-Hall代数。在本报告中，我们定义广义 Frobenius范畴与 semi-derived Ringel-Hall代数，将上述所有的构造统一在一个框架之下。这是与林记合作的工作。

In this talk, we will introduce the constant rank theorem, a technique to study the convexity of solutions to PDEs. In particular, the constant rank theorem of spacetime level sets of heat equation, a joint work with Xinan Ma and Paolo Salani, will be particularly discussed.

We will introduce the notion of free boundary constant p-mean curvature (CPMC) surface in a 3-dimensional pseudohermitian manifold with boundary. For the domain bounded by the Pansu sphere in the 3-dimensional Heisenberg group, we will talk on examples of free boundary CPMC surfaces which are rotationally symmetric about the t-axis. This is a joint work with Shujing Pan and Yongbing Zhang.