# Pure Maths Colloquium: Sean Dewar

This talk is part of the Pure Maths Colloquium at the University of St Andrews. Check out our upcoming talks at https://theran.lt/pure-colloquium/.

 Where: Zoom 818 6838 3009 When: Jan 27 2022 @ 16.00 Video: Link (internal)
 Speaker: Sean Dewar Johann Radon Institut Title: Homothetic packings of centrally symmetric convex bodies

Given $$n$$ discs with randomly selected radii, it was shown by Connelly, Gortler and Theran that there can be at most $$2n-3$$ contacts in any packing involving the $$n$$ discs. They also conjectured that the opposite is true; given a planar graph $$G$$ where every subgraph on $$m > 1$$ vertices has at most $$2m-3$$ edges, $$G$$ can be realised as the contact graph of a random disc packing with non-zero probability. I will discuss how these ideas can be extended to homothetic packings of a centrally symmetric (c.s.) convex bodies. The main results are (i) for any strictly convex and smooth c.s. convex body $$\mathcal{C}$$, every random homothetic packing of $$\mathcal{C}$$ has a $$(2,2)$$-sparse contact graph (i.e. all subgraphs of $$G$$ on $$n$$ vertices have at most $$2n-2$$ edges); (ii) for almost every c.s. convex body $$\mathcal{C}$$, we can realise all $$(2,2)$$-sparse planar graphs as the contact graph of a random homothetic packing with positive probability.