Mathematics Department Colloquium: The first hundred years of Helly’s theorem

Colloquium | October 12 | 4:10-5 p.m. | 60 Evans Hall

 Jesus De Loera, University of California, Davis

 Department of Mathematics

The classical theorem of Edouard Helly (1913) is a masterpiece of geometry. In the simplest original form it states that if a finite 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. This theorem has since found applications in many areas of mathematics, most particularly convex analysis, discrete geometry, optimization, computational geometry, number theory, algebraic geometry, etc. My lecture will begin explaining the basics and proceed with a selection of lovely applications of Helly's theorem and some of its many generalizations and variations. The last part of the talk I will present our new work about discrete versions of Helly’s theorem. This part of the story originated in the 1970’s with work of Doignon, Bell, and Scarf (arising in Economics theory). I present joint work with Aliev and Louveaux and with La Haye, Oliveros, Roldan-Pensado. I promise I will provide several open questions and students (including undergrads) are guaranteed to understand a big portion of this talk.