Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, October 15, 2009, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Uli Wagner

Complete Minors of Hypergraphs & Simplicial Complexes (Part 1. Definitions & Motivation)

Minors of graphs are a fundamental and powerful notion. Lots of things are known about them (e.g., the theorems of Kuratowski and Wagner (sadly, no relative of mine) characterizing planar graphs in terms of forbidden minors, or the monumental Robertson-Seymour graph minor theorem), and many open problems remain (Hadwiger's conjecture being maybe the most notorious of these).

How should one define minors of higher-dimensional hypergraphs or simplicial complexes?

The answer depends of course on what one wants to do with them. Our primary motivation is to study embeddability problems, for instance: Can a given 2-dimensional complex (i.e., 3-uniform hypergraph) be embedded into 4-space? And if so, how many triangles can it contain?

We discuss several possible definitions of higher-dimensional minors (in particular the notion of admissible-contraction-and-deletion minors due to Nevo) and the subtle differences between them.

The difficulty with the available notions of minors is that either they are unrelated to embeddability, or it seems (or, in some cases, simply is) impossible to prove a suitable forbidden minor theorem for them (what exactly we mean by "suitable" will be explained in the talk).

We propose a new definition, so-called homological minors (which are inspired by, but more general than, the notion proposed by Nevo).

In this first part, we give the definition, including a mini-introduction to van Kampen's obstruction embeddability, and prove some easy facts about homological minors.

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