[KAIST Discrete Math Seminar] 12/10 FRI 4PM (Xavier Goaoc, Helly numbers and nerve theorems)

[Sang-il Oum]

***** KAIST Discrete Math Seminar *****

DATE: December 10, Friday

TIME: 4PM-5PM

PLACE: E6-1, ROOM 1409

SPEAKER: Xavier Goaoc, LORIA, INRIA Nancy – Grand Est, France.

TITLE: Helly numbers and nerve theorems

http://mathsci.kaist.ac.kr/~sangil/seminar/entry/20101210/

The Helly number of a collection of sets is the size of its largest
inclusionwise minimal subfamily with empty intersection. The precise
conditions that lead to bounded Helly numbers have been studied since
the 1920′s, when Helly showed that the Helly number of any collection
of compact convex sets in R^d has Helly number at most d+1.

I will discuss a proof that any collection of subsets of R^d where the
intersection of any subfamily consists of at most r connected
components, each of which is contractible, has Helly number at most
r(d+1). I will show how this implies, in a unified manner,
quantitative bounds for several Helly-type theorems in geometric
transversal theory.

Our main ingredients are a new variant of the nerve, a “homological
nerve theorem” for this structure and an extension of a projection
theorem of Kalai and Meshulam.

This is joint work with Eric Colin de Verdiere and Gregory Ginot.