Meeting times: Wednesdays 1-3pm, starting January 20 via Microsoft teams (To register send a message to gradstud@maths.ox.ac.uk or contact me if there are any issues).

This course is hosted by the Taught Course Centre (TCC) of Bath-Bristol-Imperial-Oxford-Warwick.

Synopsis: The course is suited for students with interest in either or both probability and algebraic structures, who would like to explore the fruitful interactions between these different areas or become familiar with the actively growing field of integrable probability.

Integrable probability is a relatively new field which investigates and exploits connections between probability theory with algebraic combinatorics, representation theory and integrable systems. The motivation for the development of the set of methods that constitutes integrable probability has been the analysis of statistical mechanics models around the Kardar-Parisi-Zhang universality (longest increasing subsequences, last passage percolation, random polymers, six vertex models etc.).

In this course we will go through some of the fundamental principles, techniques and probabilistic models. Some familiarity with basic Markov process will be good. Familiarity with algebraic and combinatorial structures, although welcomed, is not necessary as we will be introducing the notions needed. Topics we will cover are:

• Overview of the Kardar-Parisi-Zhang universality and related models
• Robinson-Schensted-Knuth bijections
• Schur, Whittaker and other symmetric functions
• Cauchy identities, Pieri and branching rules
• Fredholm determinants and Tracy-Widom limits
• Markovian dynamics on arrays of particles (including Gelfand-Tsetlin patterns)
• Six-vertex model and Yang-Baxter equations

Lecture notes:

I will be mostly following these set of notes but some selection will be made and some additional material will be presented.

Classroom slides (Lectures 1, 2)

Lecture 3

Lecture 4

Lecture 5

Lecture 6

Lecture 7

Lecture 8

Additional material: A number of nice reviews on integrable probability, that cover different aspects of the field, exists including:

Lectures on Integrable Probability by Borodin-Gorin

Integrable probability: From representation theory to Macdonald processes, by Borodin-Petrov

Lectures on Integrable probability: Stochastic vertex models and symmetric functions by Borodin-Petrov