We will talk about two recent work: 1. A reaction-diffusion model with nonlocal interactions between ribbed mussels, marsh grass and sediment is proposed. Mussel-grass aggregating behavior is suggested to be a result of nonlocal density-dependent interactions between mussels. The model is a reaction-diffusion system with cooperative kinetic dynamics which is outside of the regime of classical Turing instability mechanism, but the nonlocal interaction causes the generation of spatial patterns. The nonlocal model can be approximated by a bi-harmonic PDE, which produces analytic results marching the numerical simulation of full nonlocal model. 2. A reaction-diffusion model is proposed to describe the evolution of spatial distributions of ROP1 and calcium on the pollen tube tip. The cytoplasmic ROP1 activate ROP1 on the membrane and the calcium ions inhibit ROP1, while ROP1 controls calcium influx with a time delay.

In this talk, we will present our work on open Lie group actions on open manifolds. Witten genus and elliptic genera are modular topological invariants for manifolds, which are closely related to representation of loop groups and the hypothetical index theory on free loop space as well as the elliptic cohomology theory in algebraic topology. They find applications in problems of positive curvature and group actions on manifolds. In this talk, we will briefly introduce these invariants and present our joint work with Varghese Mathai on generalizing them to open manifolds with proper actions of open Lie groups.

In this talk, we shall present our recent works on primes in arithmetic progressions with friable indices, joint with Jianya Liu and Ping Xi. Denote by $/pi(x,y;q,a)$ the number of primes $p/leqslant x$ such that $p/equiv a/bmod q$ and $(p-a)/q$ is free of prime factors larger than $y$. Assume a suitable form of Elliott--Halberstam conjecture, it is proved that $/pi(x,y;q,a)$ is asymptotic to $/rho(/log(x/q)//log y)/pi(x)//varphi(q)$ on average, subject to certain ranges of $y$ and $q$, where $/rho$ is the Dickman function. Moreover, unconditional upper bounds are also obtained via sieve methods.

著名的Green-Tao定理证明了素数中存在任意长度的算术数列。Shapiro型素数是数论中的一类特殊的、重要的素数数列，本报告将介绍Shapiro型素数中的Green-Tao定理，并给出相关的定量结果。

I shall present results obtained with Guillaume Ricotta on the distribution of paths build from partial Kloosterman sums. These results are an extension of a previous study by Kowlaski and Sawin.

The statistical multiscale method coupling with extended finite element method is presented to analyze the damage and fracture of FRP(fiber reinforced polymer) reinforced concrete structures. The FRP reinforced concrete material is composed of plain concrete and FRP reinforcement structure. Firstly, the statistical multiscale method is applied to predict the stiffness and strength parameters of plain concrete. In the FRP reinforced concrete material, the plain concrete is equivalent to homogeneous material through the homogenization process and stirrups and longitudinal reinforcement are enhancement phase. And then, the extended finite element method is introduced to analyze the four-point bending process of different FRP reinforced concrete material structure.

探索性数据分析对于建立合理的统计模型、使用恰当的统计推断方法以及验证统计模型和统计推断所需要的假设条件至关重要。对于低维数据，探索性数据分析的方法和理论发展较为成熟。但是，对于高维数据而言，发展探索性数据分析方法和理论非常困难。对高维数据进行降维，是发展探索性高维数据分析方法和理论的合理选择。高维数据降维需要解决两个非常核心的科学问题：如何刻画数据的非线性相依关系以及如何在不假设任何模型的前提下对高维数据进行降维。不管使用哪种降维方法，一个合理的要求是希望降维以后的数据还能完整包含原始高维数据所包含的信息。我们将系统讨论这两个科学问题的可能解决办法。