LIMDA Joint Seminar: Arnau Padrol
Title: Colorful simplicial depth, Minkowski sums, and generalized Gale transforms. Speaker: Arnau Padrol, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris 6).
- https://mat.upc.edu/ca/activitats/limda-joint-seminar-arnau-padrol
- LIMDA Joint Seminar: Arnau Padrol
- 2016-11-02T12:00:00+01:00
- 2016-11-02T23:59:59+01:00
- Title: Colorful simplicial depth, Minkowski sums, and generalized Gale transforms. Speaker: Arnau Padrol, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris 6).
02/11/2016 des de 12:00 (Europe/Madrid / UTC100)
Abstract
The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. (2006).
Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton (2003) in the theory of normal surfaces.
This is joint work with Adiprasito, Brinkmann, Paták, Patáková and Sanyal.