The theory of cluster algebras was set up by Fomin and Zelevinsky in a series of articles in 2001 and the next few years. In these references, they proposed some problems and conjectures on cluster algebras. In this talk, we will mention the progress of the positivity conjectures of cluster variables and d-vectors, the conjectures on sign-coherence of c-vectors and g-vectors, the problems of F-polynomials and g-vectors, the uniqueness of seed under mutation equivalence and the totality of sign-skew-symmetric matrices, the unistructurality of cluster algebras, etc.

In this talk, a brief review will be given for the insurance risk theory. Some recent results on optimal control of reinsurance, investment and dividend with various objectives will be presented. Different approaches to solve the optimality problems will be discussed and optimal strategies are analyzed. Specifically, some mean-variance optimal control problems are to be discussed in some detail.

The Ricci flow, introduced by R. Hamilton in 1982, evolves the initial geometry of a given space by the parabolic Einstein equation. One of the most important issues in the study of the Ricci flow is to understand the formation of singularities. It turns out generic singularities of the Ricci flow are essentially modeled by Ricci solitons. In this talk, I will discuss the phenomena of singularity formation in the Ricci flow and present some recent progress on Ricci solitons.

马氏过程通常是指具有马尔可夫性的随机过程，是概率论中所关心的最重要的一类随机过程，其中有许多常见的数学模型，例如 布朗运动，泊松过程等，与其它数学分支以及实际应用中都具有重要的意义，有丰富的结果。本讲座将系统地介绍马氏过程家族以及它们的构造思路。

The maximal volume entropy rigidity of Ledrappier-Wang asserts that a compact n-manifold with Ricci curvature bounded below by -(n-1) achieves the maximal volume entropy iff the manifold is hyperbolic. In this talk, we will report a recent work that if a manifold almost achieves the maximal volume entropy, then the manifold is diffeomorphic to a hyperbolic space. This is a join work with Lina Chen (ECNU) and Shicheng Xu (CNU).

We consider the optimal investment problem with probability distortion (or weighting) and general non-concave utility functions (e.g. S-shape utility). This generalizes and nests some previous literature on mathematical behavior portfolio choice, in which either the probability distortion or the non-concave utility, but not both, is considered. We propose a novel relaxation method to solve it utilizing the concave envelope of the utility function and by relaxing the probability distortion effect through concavification. We establish sufficient conditions to guarantee the existence and uniqueness of the optimal solutions. We apply our method to solve some representative problems scattered in the literature in a unified fashion, and in particular to the hedge fund profit sharing problem with probability weighting.

The VAR model involves a large number of parameters so it can suffer from the curse of dimensionality for high-dimensional time series data. The reduced-rank coefficient model can alleviate the problem but the low-rank structure along the time direction for time series models has never been considered. We rearrange the parameters in the VAR model to a tensor form, and propose a multilinear low-rank VAR model via tensor decomposition that effectively exploits the temporal and cross-sectional low-rank structure. Effectiveness of the methods is demonstrated on simulated and real data.

本报告将介绍一些关于运动策略演变的偏微分方程模型，具体内容包括单个物种模型，竞争物种模型，以及连续种群模型。我们将通过数学分析和数值模拟，试图回答什么样的空间移动策略是进化稳定的。本报告的大部分内容会适合于研究生和高年级本科生。