## ERC Advanced Grant - 2015/2021 - IntRanSt - 669306

### Title

Integrable Random Structures
### Principal Investigator

Neil O'Connell
### Summary

The aim of this project is to develop and study integrable / exactly solvable models in probability.
Examples of such models include interacting particle systems, random polymers, random
matrices and various combinatorial models related to the theory of symmetric functions
and Young tableaux. A common theme is some underlying algebraic structure which
accounts for the integrability of the model in question. The project has a strong emphasis
on models of representation-theoretic origin, particularly in the context of the theory of
Young tableaux and its vast generalisations. For example, there is a remarkable
connection between a birational version of the celebrated Robinson-Schensted-Knuth
correspondence and GL(N)-Whittaker functions; moreover, this connection may be applied
to the study of a class of random polymer models which belong to the so-called KPZ
universality class of random matrix theory. The resulting theory has been substantially developed
within the context of this project, including extensions to models with additional symmetries
and also non-commutative versions. Some other recent developments include: a new approach
to the study of moments of random matrices, connections between random matrices and SLE,
progress on random sorting networks, and scaling limits for Whittaker measures.
### Team

Jonas Arista (Sep 2015 - Sep 2020)
Elia Bisi (July 2018 - Sep 2020)
Philip Cohen (PhD student, Sep 2017 -)
Fabio Deelan Cunden (Sep 2016 - Sep 2020)
Antoine Dahlqvist
(Oct 2017 - Dec 2018)
Samuel Johnston (Oct 2017 - Sept 2019)
Jon Warren (Jan - Jun 2020)
### Activities

Probability seminars
Workshop: Randomness and Symmetry, UCD, June 18-22, 2018
Workshop: Random Matrices and Integrable Systems, UCD, May 23, 2018
Introductory lectures on random matrices and free probability, UCD, November 10 - December 8, 2017