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。

报告一：

时间：2018年1月23日14:30

地点：数计学院4号楼229室

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-Häggkvist 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.

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报告二：

时间：2018年1月25日10:30

地点：数计学院4号楼229室

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 $k\\ge 3$.

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