Final Report Summary - EC (Extremal Combinatorics)
Combinatorics forms a challenging and fundamental part of mathematics, but is in the happy position of being relatively accessible to a wider audience. One of its most exciting and rapidly developing branches is Extremal Combinatorics, which deals with finding the extremal values of a function defined on some class of combinatorial objects. This project concerned three objectives within Extremal Combinatorics, namely Hypergraph Embeddings, Generalised Turan Problems, and Intersections of Set systems. Within these objectives, the main achievements have been (i) a series of papers developing and applying a hypergraph blowup lemma and a geometric theory of hypergraph matching, (ii) two papers solving many cases of a conjecture of Erdos and Simonovits on a generalised bipartite Turan problem, (iii) the development of a new way of applying Dependent Random Choice to intersection problems. The methods developed in these papers are already finding further applications and will no doubt continue to have an impact on the research landscape.