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中山大学胡平副教授学术报告

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报告一  

报告时间:2018年1月23日14:30  

报告地点:数计学院4号楼225

报告题目:Flag Algebra Introduction

报告摘要:Razborov was awarded the 2013 David P. Robbins  Prize for introducing a new powerful method, flag algebras, to solve problems in  extremal combinatorics. This method has been applied to attack the  Caccetta-H?ggkvist conjecture, various Turán-type problems, extremal problems in  a colored environment, and also to problems in geometry. I will give an  introduction to this method by presenting small examples.


报告二  

报告时间:2018年1月25日10:30  

报告地点:数计学院4号楼229

报告题目:Flag Algebra Applications

报告摘要:I will talk about results that cannot be proved  directly from plain Flag algebra method.Let C(n) denote the maximum number of  induced copies of 5-cycles in graphson n vertices. With Balogh, Lidicky and  Pfender, we show that for n large enough we have C(n)=abcde +  C(a)+C(b)+C(c)+C(d)+C(e), where a+b+c+d+e = n and a,b,c,d,e are as equal as  possible. Moreover, for n being a power of 5, we show that the unique graph on n  vertices maximizing the number of induced 5-cycles is an iterated blow-up of a  5-cycle.

    Erdos, Faudree and Rousseau conjectured in 1992 that for every  $k\ge 2$ every graph with n vertices and $n^2/4+1$ edges contains at least  $2n^2/9$ edges that occur in $C_{2k+1}$. Very recently, Furedi and Maleki  constructed n-vertex graphs with more than $n^2/4$ edges such that only  $(2+\sqrt{2}+o(1))n^2/16\approx0.213n^2$ of them occur in $C_5$, which disproves  the conjecture for $k=2$.. With Grzesik and Volec, we use a different approach  to tackle this problem and obtain exact result. We adapt the approach to show  that the conjecture is true for $k\ge 3$.

报告人简介:胡平,本科毕业于北京大学数学系,2014年5月在美国伊利诺伊大学香槟分校(University  of Illinois at Urbana-Champaign)获得数学博士学位,2014年10月至2017年8月在英国华威大学(University of  Warwick) 任研究员(Research  Fellow),2017年入职中山大学任副教授。一直从事组合数学领域中极值组合方向的科学研究,在领域内的Ramsey理论,Turan理论,染色问题等方向均有成果。目前,在组合数学领域权威期刊JCTB、RSA、CPC、European  J Combin等发表多篇论文。研究内容是组合数学领域的前沿研究方向。芝加哥大学Razborov教授因其发明的旗代数(Flag  Algebra)工具在极值组合中的广泛应用被美国数学协会(AMS)于2014年授予Robbins奖,而胡平的主要研究方向之一为应用和拓展旗代数工具。胡平的另一研究方向是应用和拓展极限图工具。与旗代数有紧密联系的另一工具是由Lovász和Szegedy发明的极限图(Graph  limit)工具,用分析的方法研究图的极限结构,从而得到离散结构的信息。