报告人：Jürgen Herzog  德国杜伊斯堡-埃森大学教授

报告时间：2017年3月23日（周四）下午3:00-4:00

报告地点：维格堂113

报告摘要:  Vasconcelos, Villarreal and Simis are the founders of the theory edge

ideals of a graph { a theory,which has become subject of intensive research and numerous publications. Morerecently, motivated by applications in Algebraic Statistics, binomial edgeideals have been introduced in my joint paper with Hibi, Hreinsdóttir, Kahle andRauh. To each edge e={i,j}of a graph G on [n] one attachesthe binomial fe = xiyj-xjyi, and the ideal IG generated by allthese binomials.

In this lecture I will surveysome of the known results. The Gröbner bases of binomial edge ideals will be described. Graphs whosebinomial edge ideals have a quadratic Gröbner basis are called closed graphs. We present thecombinatorial characterization of closed graphs, and determine the minimalprime ideals of any binomial edge ideal in terms of cut point properties of theunderlying graph. Furhermore, the regularity,

projective dimension andCohen{Macaulay property of binomial edge ideals will be discussed with focus onthe theorems of Ene and Zarojanu, Matsuda and Murai, and of that Kiani andSaeedi Madani. Several open problems will be addressed.