In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. In several joint works with S.Seto, L. Wang; C. He; and X. Dai, S.Seto, we prove the conjecture for the sphere. Namely for any strictly convex domain in the unit $S^n$ sphere, the gap is $/ge 3/frac{/pi^2}{D^2}$. As in B. Andrews and J. Clutterbuck's work, the key is to prove a super log-concavity of the first eigenfunction.

欧几里德《几何原本》极大地影响了整个数学的发展。其核心内容是平面几何。凸几何与微分几何均始于平面几何。从历史的角度去探讨一下凸几何如何从古典的平面几何发展而来, 其与微分几何的区别与联系, 以及分析的工具所扮演的角色, 或许能在一个方面展示一点数学的美妙与方向。

题目：我的学术之路主讲人：张益唐（美籍华裔数学家）时间：2017年6月22日 13：30地点：中法中心C201（请在13：15前入场完毕）附：张益唐先生简介张益唐，美籍华裔数学家，美国加州大学圣塔芭芭拉分校教授。1955年出生于上海，1978年考入北京大学数学系，1992年毕业于美国普渡大学，获博士学位。自2013年始荣获多项国际国内大奖，包括获晨兴数学卓越成就奖、奥斯特洛夫斯基奖(Ostrowski Prize)、科尔（Cole）数论奖、罗夫·肖克...

In this talk, I will report our recent research on time-delayed compartmental population models in a periodic environment. We first establish the theory of basic reproduction ratio R0 for such systems. It is proved that R0 serves as a threshold value for the stability of the zero solution of the associated periodic linear systems. As an illustrative example, we also apply the developed theory to a periodic SEIR model with an incubation period and obtain a threshold result on its global dynamics in terms of R0.

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio R0 and establish a threshold type result on the global dynamics in terms of R0. More precisely, we show that the unique disease-free periodic solution is globally asymptotically stable if R0<1, and the model system admits a unique positive periodic solution which is globally asymptotically stable if R0>1

in this introductory talk, I will give an overview of the network sciences, and in particular, some basic metrics and models to analyze and model social networks.

For bootstrap sample sums resulting from a sequences of random variables , a very general central limit theorem is established. The random variables do not need to be independent or identically distributed or to be of any particular dependence structure. Furthermore, no conditions, including moment conditions, are imposed in general on the marginal distributions of the . As a special case of the main result, a result of Liu (1988) concerning independent but not identically distributed is extended to a larger class of parent sequences.

We generalize the P(N)-graded Lie super algebras of Martinez-Zelmanov. This generalization is not so restrictive but sufficient enough so that we are able to have a classification for this generalized P(N)-graded Lie super-algebras. The upshot is that the generalized P(N)-graded Lie super-algebra L is centrally isogenous to a matrix Lie super-algebra coordinated by an associative super-algebra with a super- involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov’s theorem for type P(N). The motivation of this generalization comes from the fermionic-bosonic module construction.