# The Chern-Simons Research Lectures

The goal of the Chern-Simons Research Lectures is to bring active, leading researchers in mathematical physics to Berkeley for a short series of lectures. Begun in 2010, the Chern-Simons Research Lectures are supported with funds from the Chern-Simons Chair in Mathematical Physics, which was established by the Simons family in honor of Jim Simons.

**The 2017 Chern-Simons Research Lectures** will be given by Ivan Corwin of Columbia University, on April 3-7, 2017.

Professor Corwin works to unify algebraic structures within mathematics, build bridges between these structures and domains of physics, and discover universal phenomena within these domains. He has uncovered universal distributions (modern day parallels of the bell curve) in models of interface growth, traffic flow, mass transport, turbulence, and shock-fronts.

He will give lectures on Monday, Wednesday, and Friday, and the Mathematics Department colloquium on Thursday.

**Stochastic Quantum Integrable Systems**

In this series of lectures I will explain how structures from quantum integrable systems can be employed to discover and analyze a variety of probabilistic systems. Studying asymptotics of these systems reveals universal behaviors which should hold true for larger universality classes

**Lecture 1:**

Monday, April 3, 2017

3:00 pm - 5:00 pm, 2 Le Conte Hall

**q-TASEP, q-Bosons and their KPZ limits**

Abstract: We will introduce q-TASEP and q-Bosons which are integrable discretizations of the KPZ stochastic PDE and delta Bose gas. Using these discretizations, we will give a derivation of the statistics describing the KPZ stochastic PDE. This is a rigorous parallel to the non-rigorous "replica method" often employed in studying KPZ directly. We will also briefly discuss some of the stochastic analysis which goes into showing convergence of q-TASEP to the KPZ equation.

**Lecture 2:**

Wednesday, April 5, 2017

2:00 pm - 4:00 pm, 325 LeConte Hall

**Bethe ansatz and Markov duality**

Abstract: We will relate q-TASEP and q-Bosons to the family of higher-spin stochastic six vertex models. We then explain how to analyze these higher spin models via Bethe ansatz and Markov dualities. In particular, we will explain the completeness and orthogonality of Bethe ansatz eigenfunctions, and further develop their properties as symmetric rational functions.

**Lecture 3:**

Friday, April 7, 2017

10:00 am - 12:00 pm, 325 LeConte Hall

**Bridges to Macdonald processes and Gibbsian line ensembles**

Abstract: Macdonald polynomials are another algebraic structure which admit meaningful probabilistic applications in the form of Macdonald processes. In this lecture we describe how the higher spin stochastic six vertex models can be related to the hierarchy of Macdonald processes through explicit formulas and through the Yang-Baxter equation. This provides an alternative way to extract asymptotics such as discussed in earlier lectures. Macdonald processes enjoy certain Gibbs structures and as an application we discuss a Gibbsian line ensemble structure enjoyed by the stochastic six vertex model and by ASEP which enables us to prove the conjectural 2/3 transversal exponent for these models.