From May 27 to June 8, 2019, Department of Mathematics of University of Virginia organizes a two-week summer school in Integrable Probability. 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.

The school will have 4 mini-courses:

- Dmitry Chelkak (École Normale Supérieure, Paris, France)
**Planar Ising model: from combinatorics to CFT and s-embeddings**In theoretical physics, the critical planar Ising model serves as a toy example, in which many precursors of Conformal Field Theory objects and structures exist and can be studied directly in discrete, before passing to the small mesh size limit. Mathematically, a number of results on convergence and conformal invariance of such limits were established during the last decade, both for correlation functions and for interfaces (domain walls) arising in the model. In this mini-course we plan to discuss- discrete fermions and the Kadanoff-Ceva spin-disorder formalism - crucial tools that allow one to analyse the planar Ising model;
- streamlined version of the classical computation of the magnetization via orthogonal polynomials;
- results on convergence of critical correlation functions (energy densities, spins, ...) in bounded domains to CFT limits;
- recent ideas on appropriate embeddings of weighted planar graphs that play the same role for the planar Ising model as Tutte’s barycentric embeddings do for random walks, allowing one to use discrete complex analysis techniques beyond "regular" lattices.

- Ole Warnaar (University of Queensland, Brisbane, Australia)
**Schur functions and Schur processes**Abstract TBA - Tomohiro Sasamoto (Tokyo Institute of Technology, Tokyo, Japan)
**Fluctuations of 1D exclusion processes: exact analysis and hydrodynamic approach**One dimensional exclusion processes are stochastic processes in which many particles perform random walks under exclusion constraint. They have been playing important role in the fields of stochastic interacting systems in probability theory and non-equilibrium statistical mechanics in physics. For the last two decades, fluctuations of the processes have been studied quite intensively, since the seminal work by Johansson[1-1] on totally asymmetric simple exclusion process (TASEP) showing that the current fluctuation of TASEP with step initial condition is described by the GUE Tracy-Widom distribution. There have been a vast accumulation of generalizations and related results, but there are still many intriguing questions and problems to be solved.

In these lectures, we discuss a few new directions in the studies of fluctuations of exclusion processes. We also stress that such studies provide valuable insight to other methods based on hydrodynamic ideas which can be applied to a wider class of interacting particle systems. In the first lecture we review the basics of the subject. After introducing a few models such as the asymmetric simple exclusion process(ASEP) and the Kardar-Parisi-Zhang (KPZ) equation, we explain how one can study their fluctuations for the case of TASEP[1-2]. In the second lecture, we show that an approach introduced in [2] using Frobenius determinant can be applied to a large class of models in a unified manner. In the third lecture we explain our recent result on a two-species exclusion process and connection to the nonlinear fluctuating hydrodynamics[3]. In the last lecture we will consider an application of the techniques to study the large derivation in the symmetric exclusion process[4-1,2].

**References**- [1-1] K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. (2009) 437-476. [arXiv:math/9903134]
- [1-2] T. Sasamoto, Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques, J. Stat. Mech. (2007) P07007. [arXiv:0705.2942]
- [2] T. Imamura, T. Sasamoto, Fluctuations for stationary q- TASEP, to appear in Prob. Th. Rel. Fields. [arXiv:1701.05991]
- [3] Z. Chen, J. de Gier, I. Hiki, T. Sasamoto, Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process, Phys. Rev. Lett. 120, 240601 (2018). [arXiv:1803.06829]
- [4-1] T. Imamura, K. Mallick, T. Sasamoto, Large deviations of a tracer in the symmetric exclusion process, Phys. Rev. Lett. 118, 160601 (2017). [arXiv:1701.05991]
- [4-2] T. Imamura, K. Mallick, T. Sasamoto, Distribution of a tagged particle position in the one-dimensional symmetric simple exclusion process with two-sided Bernoulli initial condition, arXiv:1810.06131.

- Paul Zinn-Justin (University of Melbourne, Melbourne, Australia)
**Quantum integrability and symmetric polynomials**Abstract TBA

Talks are in Clark 107 (schedule)

The summer school is supported by the National Science Foundation Focused Research Group grant (DMS-1664617)

Organizers: Leo Petrov, Axel Saenz

Scientific Committee: Jinho Baik, Alexei Borodin, Ivan Corwin, Vadim Gorin, Leo Petrov