Content: In the past decade there has been an explosion of interest on the interface and relations between symmetric functions and probability. This has led, on the one hand, to remarkable resolutions of open problems in statistical mechanics and, on the other, to progress in the algebraic counterparts. The whole interaction has created a new and very active field named Integrable Probability.

This module will inform 4th year students on these novel developments and to equip them with the foundations on symmetric functions and probability that are required to enter this new field.

On the symmetric function side we will look into Schur functions and their various Macdonald generalisations. We will also look at the interplay with algebraic combinatorics and discuss Young tableaux and the Robinson-Schensted-Knuth correspondence. Connections with Crystal Theory of Kashiwara and Lustig will also be made as well as other fundamental notions from representation theory.

On the probability side we will see how the algebraic tools are combined with probabilistic notions to attack problems of random growth, in particular related to models in the Kardar-Parisi-Zhang (KPZ) universality. Other statistical mechanics models such as Ice and Vertex models will be discussed. We will explore how the interplay of probability and algebra leads to asymptotic analysis of the statistical of these models and their relation to statistics from Random Matrix Theory. We will also show how the probabilistic intuition leads to important algebraic identities such as Cauchy and Littlewood identities.

Indicative syllabus:

1. Basic symmetric functions
2. Schur functions
3. Young tableaux and Robinson-Schensted-Knuth correspondence
4. Macdonald functions
5. KPZ universality and integrable probabilistic models
6. Asymptotics analysis and Random Matrix (ie Tracy-WIdom) asymptotic laws

Learning Outcomes: By the end of the module students will:

1. Understand the interplay of probability and algebra
2. Introduce basic tool from algebraic combinatorics
3. Introduce symmetric functions and the understand their representation theoretic and combinatorial nature
4. Introduce probabilistic models in the KPZ universality and other related integrable models from statistical mechanics
5. Set the foundations, based on the above interplay, to show asymptotic probabilistic laws of KPZ and statistical mechanics models and their relation to Random Matrix Theory

Books: TBC