# 11月29日 彭岳建教授学术报告（数学与统计学院）

彭岳建教授2001年于美国埃默里大学（Emory University）获得理学博士学位。2002-2012年在美国印第安纳州立大学（Indiana State University）历任助理教授、副教授、教授（终身）。2012年作为“湖南省百人计划”特聘教授回到湖南大学。目前，彭岳建教授已经在极值组合与图论及相关领域做出了许多出色的工作，在国际组合图论权威刊物JCTB、JCTA、CPC、JNT（数论杂志）等发表论文40多篇。一直得到国家自然科学基金的资助。

Given a positive integer $n$ and an$r$-uniform hypergraph $H$, the {\em Tur\'an number} $ex(n, H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices. The {\emTur\'{a}n density} of $H$ is defined as $\pi(H)=\lim_{n\rightarrow\infty} {ex(n,H) \over {n \choose r } }.$ The {\em Lagrangian density } of an $r$-uniform graph $H$ is $\pi_{\lambda}(H)=\sup \{r!\lambda(G):G\;is\;H\text{-}free\}$, where $\lambda(G)$ is the Lagrangian of $G$.  The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems.  Recently, Lagrangian densities of hypergraphs and Tur\'{a}n numbers of their extensions have been studied actively.

The Lagrangian density of an $r$-uniform hypergraph $H$ is the same as the Tur\'{a}n density of the extension of $H$. Therefore, these two densities of $H$ equal if every pair of vertices of  $H$  is contained in an edge. For example, to determine the Lagrangian density of $K_4^{3}$ is equivalent to determine the Tur\'an density of $K_4^{3}$. For an $r$-uniform graph $H$ on $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$, where $K_{t-1}^r$ is the complete $r$-uniform graph on $t-1$ vertices. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is $\lambda$-perfect if $\pi_{\lambda}(H)=r!\lambda{(K_{t-1}^r)}$.  A result of Motzkin and Straus implies that all graphs are $\lambda$-perfect.  It is interesting to explore what kind of hypergraphs are $\lambda$-perfect. We present some open problems and recent results.  / 