Presented by: 
Éric Colin de Verdière (ENS Ulm, Paris)
Date: 
Tue 3 Jul, 3:00 pm - 4:00 pm
Venue: 
Room 442, Building 67

A family F of sets has Helly number at most k if the following holds: If any subfamily of F of cardinality at most k has non-empty intersection, then F has non-empty intersection.  Helly's well-known theorem from convex geometry, proved in 1923, asserts that every finite family of convex sets in R^d has Helly number at most d+1.  This can be generalized: Helly himself proved in 1930 that the result holds for good covers (finite families of open sets where the intersection of each subfamily is either empty or contractible).

We further generalize this result by bounding the Helly number of acyclic families, namely, finite families of open sets where the intersection of every subfamily is a disjoint union of homology cells. As an application of this theorem, we reprove in a unified way several bounds on transversal Helly numbers in geometric transversal theory.
 

This talk will give the main ideas for proving these results; the new Helly-type theorem relies on (1) a generalization of the notion of nerve, the multinerve, a combinatorial object that has the same topology as the union of the objects in the family, if that family is acyclic, and (2) a generalization of a result by Kalai and Meshulam from 2008 on Leray numbers.

This is joint work with Grégory Ginot and Xavier Goaoc.  Preprint: arXiv:1101.6006