In combustion theory, the propagation of premixed flames is usually described by the conventional thermal-diffusional model with standard Arrhenius kinetics. Formal asymptotic methods based on large activation energy have allowed simpler descriptions, especially when the thin flame zone is replaced by a free interface, called the flame front, which separates burned and unburned gases. At the flame front, the temperature and mass fraction gradients are discontinuous.

We get estimates on the higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, in terms of the eigenvalues of the Laplace–Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result (Alexandrov, 2007) of Alexandrov.

Roughly speaking, the geometric group theory studies groups from the geometric point of view. We will talk about lengths, curvatures, areas on groups. In particular, we will introduce the notions of CAT(0) groups, Gromov hyperbolic groups and some problems.

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.