Alfredo Hubard 

From crossings to areas and back.

This talk will center on an analogue of Heilbronn's triangle problem for simple topological complete graphs. These are drawings of the complete graph in the plane in which every pair of edges intersect at most once. The motivating question is whether there exists a function t(n) that goes to 0 when n goes to infinity, such that for any simple complete topological graph contained in a region of area 1, there exists a triangle of area at most t(n). I will give an answer to this question based on recent joint work with Andrew Suk. I will  put it in context with other results in quantitative topology in which areas of regions, and intersections between faces mingle.