Random polytopes: two special seminars: Williamson & Feng

'The diameter of polytopes and the Hirsch conjecture'

Speaker: Geordie Williamson, University of Sydney

Time: 9 am - 10 am   

Abstract: Can one reasonably bound the diameter of the graph of a simple polytope? This is a fascinating open problem in combinatorial geometry. I’ll outline what is known, and in particular sketch Santos’ remarkable 2012 counter-example to the 50 year-old Hirsch conjecture. With the DeepMind team, we recently attacked this problem using machine learning. We made some progress, but this is not the focus of the talk. I want to communicate a simple problem where the right idea could be revolutionary.

'A short survey on random polytopes'

Speaker: Renjie Feng, University of Sydney

Time: 10 am - 11 am 

Abstract: Suppose we take random points in the plane from a Gaussian and look at the polytope they generate. What can I say about it? What is its expected volume, number of vertices, number of edges etc? This talk will provide an introduction to these kinds of questions. The focus is on many points in a fixed dimension, but some striking results where one allows the dimension to grow will also be touched upon.