Intensive Lecture Series

2025 January Jiwoon Park, TBA

TBA

2024 October - November Francesco Caravenna, Rongfeng Sun & Nikos Zygouras, The critical 2d Stochastic Heat Flow and related models


[2024/10/15, 16:00] Lecture 1: Overview and main results  **Slides**

We start with an overview, providing motivation and presenting the main models and results that will be discussed in this lecture series. We then describe a key link between the solutions of the Stochastic Heat Equation (SHE) and Kardar-Parisi-Zhang Equation (KPZ) and the partition function of the model of Directed Polymer in Random Environment (DP), which is central to what follows.


[2024/10/22, 16:00] Lecture 2: Polynomial chaos, sub-criticality, and disorder relevance  **Slides**

We present the important tool of "polynomial chaos", which yields explicit expansions in L^2 for our models, making a link with the notions of "sub-criticality" (in the framework of singular PDEs) and "disorder relevance" (in the framework of disordered systems). We then discuss the scaling limits of the SHE in the spatial dimension d=1, presenting key techniques of independent interest such as Lindeberg principles and hypercontractivity.


[2024/10/30, 10:30] Lecture 3: Critical dimension and phase transition  **Slides**

We focus on the critical dimension d=2. We start with a variance computation for the solution of the SHE and the DP partition function, showing the need of renormalising the disorder strength and the emergence of a phase transition. We then focus on the regime below the critical disorder strength, discussing the asymptotic log-normality of the solution.


[2024/11/06, 10:30] Lecture 4: Edwards-Wilkinson fluctuations  **Slides**

We look at the solution of the SHE or the DP partition functions in the critical dimension d=2 and below the critical disorder strength. We show that, as random fields in space, they satisfy Edwards-Wilkinson (EW) fluctuations, that is, convergence in distribution under suitable rescaling toward a log-correlated Gaussian field (solution of the EW equation, aka additive SHE). We then discuss an analogous result for the solution of the KPZ equation. Basic tools are hypercontractivity and concentration of measure.


[2024/11/14, 17:00] Lecture 5: The critical 2d Stochastic Heat Flow: outline  **Slides**

We focus on the regime of critical disorder strength in the critical dimension d=2, presenting the construction of the critical 2d Stochastic Heat Flow (SHF) and its main properties. We outline the key ingredients of the proof with a focus on coarse graining techniques. We then present an application of a refined Lindeberg principle which proves a form of universality for the SHF.


[2024/11/22, 17:00] Lecture 6: The critical 2d Stochastic Heat Flow: tools  **Slides**

We discuss some key tools for the construction and study of the SHF. We first present sharp second moment computations for the solution of the SHE or DP partition function in the critical regime, based on a renewal theory approach which leads to the so-called Dickman subordinator. We then discuss higher moment bounds based on a functional analytic approach. We conclude with a summary and a discussion of open problems.

2024 September - October Jaehwan Choi & Jinsol Seo, An Introduction to SPDEs 


In this lecture series, we will focus on the Parabolic Anderson Model (PAM). The primary goal is to understand the Sobolev regularity theory for PAM driven by space-time white noise in the sub-critical regime. We will begin by introducing the concept of space-time white noise. 


[2024/09/19, 10:00] Lecture 1: The Concept of Random Noise

Abstract: In this lecture, we will introduce the concept of Gaussian fields, including space-time white noise and colored noise. The goal is to understand the series representation of space-time white noise, which enables the use of Itô's stochastic calculus.


[2024/09/26, 10:00] Lecture 2: Sobolev Regularity Theory for the Stochastic Heat Equation 

Abstract: In this lecture, we will introduce an overview of the stochastic heat equation's regularity results. The goal is to drive the simplest L_2 regularity results and understand some generalizations containing L_p regularity results.


[2024/10/01, 10:00] Lecture 3: Sobolev Regularity Theory for the Parabolic Anderson Model  

Abstract: In this lecture, we will introduce the Sobolev regularity theory for the Parabolic Anderson Model with the space-time white noise in the sub-critical regime. Additionally, we will present similar results for generalized versions of the Parabolic Anderson Model.


[2024/10/8, 10:00] Lecture 4: Various topics in the stochastic heat equation

Abstract: This lecture will introduce various problems for the stochastic heat equation. The first is the domain problem for the stochastic heat equation, and the second is fractional generalizations of the classical heat and the stochastic heat equations.

2024 April Benjamin Gess & Daniel Heydecker, Large deviations from porous media equations and gradient flow structures  

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.

2024 January Daniel Remenik, An introduction to the KPZ universality class (online, slides)

 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.