On Hundred years of Helly theorem
主 题: On Hundred years of Helly theorem
报告人: Professor Jesus De Loera (UC Davis)
时 间: 2017-06-20 16:00-17:00
地 点: 理科1号楼1569
Abstract: Convex geometry has experienced a renaissance due to applications in optimization algorithms and machine learning.
The classical theorem of Edouard Helly (1913) is a masterpiece of convex geometry. It states that if a family $\Gamma$ of convex sets in $R^n$ has the property that every $n+1$ of the sets have a non-empty intersection, then all the convex sets must intersect. Helly's theorem has since found applications in many domains, more recently in convex discrete optimization and in computer algebra via sampling style algorithms. My lecture will begin explaining the basics of convex geometry and proceed with a selection of lovely applications of Helly's theorem. The last part of my talk will deal with some surprising new combinatorial generalizations, my favorite one is our new version of Helly’s theorem where the intersection(s) count lattice points.
This side of the story originated in the 1970's work of Doignon, Bell, and Scarf (arising in Economics theory) and was followed in joint work with Aliev and Louveaux in the last two years. I promise I will mention the history of the subject and provide some open questions. Even undegraduate students are guaranteed to understand a big portion of this talk.