In a list of problems published in 2001, Alan Sokal conjectured a value D where the independence polynomial of graphs is zero-free in a neighborhood of the complex plane around an interval of the real segment [0,D]. It is important to note that D is calculated (explicitly) using only the maximum degree of the graph.

Very recently, a proof of the validity and optimality of this conjecture was found using techniques in complex dynamics by Peters and Regts. After introducing some fundamentals, I will give a brief outline of the proof followed by a short example of the importance of using dynamics to make the problem more approachable. Time permitting, I will discuss some of the applications of the conjecture to computational complexity theory as well as physics. Speaker(s): Anthony Della Pella (University of Michigan)