Extremal and Probabilistic Combinatorics

Date: 2 - 6 June 2014

Location: Mathematical Institute, University of Oxford

Extremal combinatorics and probabilistic combinatorics are two of the most central branches of modern combinatorics. Extremal combinatorics deals with problems of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects satisfying certain requirements. Such problems are often related to other areas including computer science, logic, number theory and geometry. Probabilistic combinatorics can be described informally as a hybrid between combinatorics and probability, where the main object of study is probability distributions on discrete structures, a notable example being the theory of random graphs.

Recent years have seen some truly spectacular developments in both subjects whose impact is likely to go far beyond their immediate context. They include: the transference principle and its applications to extremal problems on random structures; recent breakthroughs on classical questions in graph Ramsey theory; the development of flag algebras as a computational approach to extremal questions; analysis of new models of random objects, partly motivated by practical applications and settings; the polynomial method and its applications to classical questions in combinatorial geometry. However, these are only a few representative examples of a healthy and burgeoning field. 

This workshop will focus on the interactions between various areas of extremal and probabilistic combinatorics. The topics will include extremal problems for graphs and set systems, Ramsey theory, combinatorial number theory, combinatorial geometry, random graphs, probabilistic methods and graph limits. We plan to bring together researchers in the field from around the world, aiming to facilitate the exchange of the latest ideas and developments and hoping to achieve substantial breakthroughs already at the workshop or soon after it. In order to promote the field and to ensure its vitality, we also plan to bring several of the brightest and most promising researchers from the younger generation.

Image by Loasa, via Wikimedia Commons