The cluster expansion is a classical tool from statistical physics used to study the phase diagram of interacting particle systems. Recently, the cluster expansion has seen a number of applications in combinatorics and the field of approximate counting/sampling. In this article, we give an introduction to the cluster expansion and survey some of these recent developments.

The slice rank polynomial method was introduced by Tao in 2016 following the breakthrough of Ellenberg and Gijswijt on the famous Cap-Set Problem, which in turn was building on work of Croot, Lev and Pach. This survey gives an introduction to the slice rank polynomial method, shows some of its early applications, and discusses the developments since then.

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh–Katz’s formula μk(Μ) = degΜ(αr–k βk) for the coefficients of the reduced characteristic polynomial of a matroid M as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes α and β in the Chow ring of Μ. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids.

Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory. This survey was prepared for the Clay Lecture to be delivered at the 2024 British Combinatorics Conference.

About twenty years ago, Green wrote a survey article on the utility of looking at toy versions over finite fields of problems in additive combinatorics. This survey was extremely influential, and the rapid development of additive combinatorics necessitated a follow-up article ten years later, which was written by Wolf. Since the publication of Wolf’s article, an immense amount of progress has been made on several central open problems in additive combinatorics in both the finite field model and integer settings. This survey covers some of the most significant results of the past ten years and suggests future directions.

Network flows over time are a fascinating generalization of classical (static) network flows, introducing an element of time. They naturally model problems where travel and transmission are not instantaneous and flow may vary over time. Not surprisingly, flow over time problems turn out to be more challenging to solve than their static counterparts. In this survey, we mainly focus on the efficient computation of transshipments over time in networks with several source and sink nodes with given supplies and demands, which is arguably the most difficult flow over time problem that can still be solved in polynomial time.