A 0/1 polytope is the convex hull of a set of 0/1 d-dimensional vectors. A conjecture of Milena Mihail and Umesh Vazirani says that the graph of vertices and edges of every 0/1 polytope is highly connected. Specifically, it states that the edge expansion of the graph of every 0/1 polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a 0/1 polytope in R^d is greater than 1 over some polynomial function of d. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random 0/1 polytope in R^d is at least 1/12d with high probability. This is joint work with Brett Leroux.

In various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability at least 5/8, must be abelian. For infinite groups, one needs to work a bit harder to define the probability that a given law holds. One natural way is by sampling a random element uniformly from the r-ball in the Cayley graph and taking r to infinity; another way is by sampling elements using random walks. It was asked by Amir, Blachar, Gerasimova, and Kozma whether a law that holds with probability 1, must actually hold globally, for all elements. In a recent joint work with Be’eri Greenfeld, we give a negative answer to their question.

In this talk I will give an introduction to probabilistic group laws and present a finitely generated group that satisfies the law x^p=1 with probability 1, but yet admits no group law that holds for all elements.

Percolation clusters induce an intrinsic graph distance called the chemical distance. Besides its mathematical appeal, the chemical distance is connected to the study of random walks on critical percolation clusters. In this talk, I will begin with a brief survey on the chemical distance. Then, I will zoom in to the progress and challenges in the 2d critical regime. A portion of this talk is based on joint work with Philippe Sosoe.