3:30 pm, Seminar Hall
Helly-type theorems and topological combinatorics
Université Paris-Est Marne-la-Vallée.
To study or manipulate a geometric object, it is sometimes useful to consider an associated combinatorial structure, which emphasizes some of its properties. I will illustrate this idea on the analysis of the intersection patterns of subsets of the space. By a theorem of Helly, if four convex sets in the plane intersect triple-wise, then they must have a point in common. This property gives rise to many generalizations, relaxing for example the assumption of convexity. I will discuss how some techniques from topological combinatorics (embeddings of graphs or simplicial complexes, homology of nerve complexes and nerve theorem) allow to unify many of these generalizations.