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Integrable Random Structures

Periodic Reporting for period 4 - IntRanSt (Integrable Random Structures)

Reporting period: 2019-03-01 to 2020-08-31

This project aims 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.

The proposed research will be of interest to a growing community of researchers working at the interface of probability, integrable systems, algebra, analysis and mathematical physics, including researchers working on integrable systems, random matrix theory, combinatorics, representation theory, tropical geometry, interacting particle systems, random polymers and the KPZ equation. As it is a very topical research area at the interface of several different areas, this research will also be of interest to a wider community of researchers in the mathematical sciences. It will also generate new research directions for younger researchers, especially the PDRAs appointed to the program, in this rapidly expanding field.

This project will support the European research community’s drive to be among the prime generators of top-quality research. The nature of pure mathematical research is that it is traditionally somewhat removed from direct commercial usage. However, we might expect that the quality and cutting-edge nature of this proposal will ultimately have far wider implications which are harder to predict at the moment. We would expect there to be a significant impact, but only over the longer term. The fields of probability, analysis, integrable systems, representation theory and statistical physics play a fundamental part in our understanding of the world and have had proven impact on every branch of science, engineering and social sciences. Probability in particular is used everywhere: medicine, the weather, demographics, economics, the stock market, communications, cell biology, particle physics, climate change, etc. This project will provide leadership to the scientific base of this subject and ensure continued impact.
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.
This is a rapidly developing field with interactions across several areas of mathematics and has already had a significant impact, as evidenced by the growing number of scientific meetings devoted to these topics. The research carried out in this project is at the cutting edge of these developments, and therefore the expected potential impact is high.