Random Matrix Theory has been of central importance in Mathematical Physics for over 50 years. It has deep connections with many other areas of Mathematics and a remarkably wide range of applications. In 2012, a new avenue of research was initiated linking Random Matrix Theory to the highly active area of Probability Theory concerned with the extreme values of logarithmically correlated Gaussian fields, such as the branching random walk and the two-dimensional Gaussian Free Field. This connects the extreme value statistics of the characteristic polynomials of random matrices asymptotically to those of the Gaussian fields in question, allowing some important and long-standing open questions to be addressed for the first time. It has led to a flurry of recent activity and significant progress towards proving some of the main conjectures. A remarkable discovery has been that the characteristic polynomials of random matrices exhibit, asymptotically, a hierarchical branching/tree structure like that of the branching random walk. However, many important questions remain open. Our aim is to attack some of these problems using ideas and techniques that have so far not been applied to them, including the theory of integrable systems, representation theory, and enumerative combinatorics. We also hope to extend these ideas to areas connected with Random Matrix Theory, including, in particular, Number Theory.

This research programme is generously funded by an Advanced Grant from the European Research Council. Previous work by members of the team in this area has been supported by a Royal Society Leverhulme Senior Research Fellowship, a Royal Society Wolfson Research Merit Award, and by the Heilbronn Institute for Mathematical Research.