Intensive Lecture Series

 TBA

 In this lecture series we explore the fluctuations around the porous medium equation, examining their connections to large deviations principles and gradient flow structures. The first part, will be a general introduction to fluctuations in conservative systems, large deviations, and gradient flow structures. In the second part, we present the full realization of this program in the context of the porous medium equation. This includes the proof of a full large deviations principle for a rescaled zero range process, and the link to a formal gradient flow interpretation of the porous medium equation by deducing a De Giorgi entropy dissipation principle from the large deviations and reversibility. In the third part of the lecture, we discuss associated degenerate PDEs with irregular coefficients, and the proof of their well-posedness by means of a kinetic theory and novel commutator estimates.

 04/11, Thu 11:00 &  04/11, Thu 14:00 &  04/12, Fri 11:00, 129-104

 The one-dimensional Kardar-Parisi-Zhang (KPZ) universality class is a broad collection of models including one-dimensional random interface growth, directed polymers and particle systems. At its center lies the KPZ fixed point, a scaling invariant Markov process which governs the asymptotic fluctuations of all models in the class, and which contains all of the rich fluctuation behavior seen there.

 In these lectures I will introduce these topics and explain how one can derive explicit formulas for the transition probabilities of the KPZ fixed point, which lead to connections with random matrix theory and with some classical integrable systems. As a starting point in the derivation we will use one of the basic models in the class, the polynuclear growth model (PNG). This is a model for crystal growth in one dimension, which is intimately connected to the classical longest increasing subsequence problem for a uniformly random permutation. The solution will be obtained through a mix of probabilistic and integrable methods.

What follows is an approximate plan for each of the lectures:

[2024/01/02, 10:00] Lecture 1: Introduction 

In this lecture I will provide a general introduction to the KPZ universality class and discuss some of its basic models. Among them is the famous KPZ equation, an SPDE which is sometimes considered to be the canonical model of random interface growth. Using this equation, I will provide a formal argument to derive the characteristic 1:2:3 scaling exponents predicted to hold for all models in the class. I will then present the KPZ universality conjecture, which states that, under this 1:2:3 rescaling, all models in the class converge to a single, universal limit, the KPZ fixed point.

[2024/01/04, 10:00] Lecture 2: Some history and some limiting objects

This lecture will be devoted to reviewing some of the relevant history of results obtained for models in the KPZ universality class, together with a rough sketch of the type of arguments that have been used to obtain them. Along the way we will define some of the first limiting processes which were derived for models in the class, including connections with distributions coming from random matrix theory.

[2024/01/09, 10:00] Lecture 3: The polynuclear growth model

In this lecture I will introduce the PNG model, and its connection with the classical longest increasing subsequence problem for uniformly random permutations. I will present explicit formulas for its transition probabilities, which are given as Fredholm determinants of a kernel which is built out of hitting probabilities of a continuous time simple random walk, the invariant measure of the process. I will also explain how these formulas lead to a connection with a classical integrable system of differential equations, the 2D Toda lattice.

[2024/01/16, 10:00] Lecture 4: Solving PNG

This lecture will be devoted to proving the Fredholm determinant formula for the PNG transition probabilities. The proof will be based on probabilistic properties of the model together with some algebraic simplifications which are inherited from its integrable structure.

[2024/01/18, 10:00] Lecture 5: From PNG to the KPZ fixed point

In this lecture I will explain how one can compute the limit of the PNG formulas to obtain the scaling limit of the model, the KPZ fixed point, and I will discuss the limiting kernels which appear in the formulas for this provess. 

[2024/01/25, 10:00] Lecture 6: Further properties of the KPZ fixed point 

This lecture will be devoted to discussing many of the main properties of the KPZ fixed point. In particular, we will see how the explicit Fredholm determinant formula derived in earlier lectures yield a connection between the KPZ fixed point and the Kadomtsev-Petviashvili (KP) equation, a famous integrable dispersive PDE, and how from this relation the connection between KPZ models and distributions from random matrix theory follows. Time permitting, we will also discuss a variational representation of the KPZ fixed point which expresses it in terms of an object known as the directed landscape.