Gyula Karolyi:
 Old and new applications of the Combinatorial Nullstellensatz in geometry

Abstract: In previous talks at this seminar Zoltán Lóránt Nagy and Péter Maga discussed applications of the Combinatorial Nullstellensatz in graph theory and additive combinatorics. Here we mostly focus on applications in finite and discrete convex geometry. An almost cover of a discrete set of points is a collection of hyperplanes that cover all points except one. After showing the short proofs of some classical results, such as Jamison's theorem and the Alon-Füredi theorem, I will present some new results obtained with Gábor Hegedüs. Beacuse of its relevance, at some point I may make a short detour to additive combinatorics as well. I will also propose some open problems in the hope that some people visiting the Erdős Center  will find them appealing.