Monday, July 8 to Friday, July 19, 2024
University of Virginia, Charlottesville, VA
The aim of the school is to educate the participants in recent trends around Integrable Probability - a rapidly developing field at the interface of probability / mathematical physics / statistical physics on the one hand, and representation theory / integrable systems on the other
Organizers: Leonid Petrov, Daniel Slonim, Mikhail Tikhonov
E-mail: lenia.petrov+vipss@gmail.com
Previous summer school (2019)
Week 1 (July 8-12)
1. Positivity Everywhere• [COURSE PAGE] • Natasha Blitvic (QMUL), TA:
Slim Kammoun (ENS)
There are many notions of positivity in mathematics, each opening up a window on a different area of current mathematics research. We will focus on the (extensive) interface between positivity and combinatorics, particularly on the algebraic or combinatorial structures that can be seen to "underpin" probability. Examples, techniques, and exercises will be drawn from integrable systems, noncommutative probability theories tied to specific types of combinatorial structure, and the modern takes on the classical moment problems.
2. The KPZ fixed point • [COURSE PAGE] • Jeremy Quastel (Toronto), TA: Xincheng Zhang (Toronto)
This course will be an introduction to KPZ universal distributions through a special integrable discretization, the polynuclear growth model.
Week 2 (July 15-19)
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3. Invariant Measures for Exclusion Processes • [COURSE PAGE] • Dominik Schmid (Bonn and Columbia), TA: Zongrui Yang (Columbia)
In this mini-course, we investigate invariant measures for exclusion processes. After a brief introduction to some classical results on the set of invariant measures for the simple exclusion processes on infinite graphs, we turn our focus to the asymmetric simple exclusion process with open boundaries, also called the open ASEP. We introduction the matrix product ansatz, and show how it can be used to study the stationary distribution of the open ASEP. We discuss different representations of the matrix product ansatz. Using a particular solution for the matrix product ansatz with Askey-Wilson polynomials, we then introduce (signed) Askey-Wilson measures, and explain how the stationary distribution of the open ASEP can be used to construct stationary solutions to the open Kardar-Parisi-Zhang equation. In the last lecture, we focus on some very recent developments for the special case of the open TASEP: a so-called two-layer Gibbs representation of its invariant measure.
4. Dimers and Embeddings • [COURSE PAGE] • Marianna Russkikh (Notre Dame), TA: Matthew Nicoletti (MIT)
Dimer model is a probability measure on perfect matchings of a graph. We present several tools to study the dimer model on planar bipartite graphs: Kasteleyn's theory, combinatorial correspondences with other models, discrete complex analysis, and embeddings of planar graphs.
Monday, Tuesday, and Thursday
- 8:00 AM - 9:00 AM • Breakfast and coffee (in front of the lecture room)
- 9:00 AM - 10:00 AM • Lecture
- 10:00 AM - 10:30 AM • Coffee Break
- 10:30 AM - 12:00 PM • Problem Session
- 12:00 PM - 2:00 PM • Lunch Break (on your own except Mondays and Tuesdays)
- 2:00 PM - 3:00 PM • Lecture
- 3:00 PM - 3:30 PM • Coffee Break
- 3:30 PM - 5:00 PM • Problem Session
Wednesday and Friday
- 8:00 AM - 9:00 AM • Breakfast and coffee (in front of the lecture room)
- 9:00 AM - 10:00 AM • Lecture
- 10:00 AM - 10:30 AM • Coffee Break
- 10:30 AM - 11:30 AM • Participant Talks / AI Discussion (Jul 12)
- 11:30 AM - 11:35 PM • Coffee Break
- 11:35 PM - 1:00 PM • Problem Session
- 1:00 PM • Lunch (on your own) and free afternoon
Registration, financial support, and accommodation deadlines
- To request University of Virginia residence hall accommodation: March 15, 2024
- To request financial support: May 1, 2024
- General registration: July 1, 2024