Intensive Lecture Series

• TBA

• TBA

• Lecture 1: 05/19, Mon 17:00 

• Lecture 2: 05/26, Mon 16:00   

• Lecture 3: 06/02, Mon 16:00   

• Lecture 4: 06/09, Mon 16:00   

** Lecture Note **

Abstract: In this mini-course, we discuss the renormalisation group method applied to the phi4 model. In Part I, the phi4 model is introduced as a smoothing of the Ising model. They are understood in the framework of statistical field theories (SFTs), and we give a brief introduction of the renormalisation group (RG) analysis of SFTs. When the dimension is greater or equal to 4, the RG argument gives an alternative characterisation of the critical point and allows precise computation of critical exponents. In Part II, we introduce the polymer expansion, which is an algebraic structure that can be used to implement a rigorous renormalisation group method. The polymer expansion can be understood as a multi-scale extension of the cluster expansion. This algebraic structure has a sufficient amount of flexibility that allows various operations that define an RG step, while it is restrictive enough to conserve the key features of the RG flow. In Part III, we introduce the key estimates that guarantee the stability of the RG flow. There are two main mechanisms. The first is the contraction due to the localisation process, and the second is the contraction on large sets. These two mechanisms are observed universally in various different rigorous implementations of the RG method. Along with the two contraction estimates, the critical point can be constructed using a fixed point argument. In Part IV, we discuss generations to higher dimensions and RG with observables. They require a generalised theory of localisation and observable fields, but the backbone of the RG remains untouched. They allow computation of multipoint functions and scaling limits.

• Lecture 1: 04/14, Mon 17:00

• Lecture 2: 04/21, Mon 17:00   

• Lecture 3: 04/28, Mon 17:00 

[2025/04/14, 17:00] Lecture 1 slide

Abstract: In this first lecture, we will provide an overview of spin glass theory, focusing on its historical development and explaining why it is significant. We will introduce the basic concepts of spin glass theory, such as the Gibbs measure, the Parisi formula, Replica Symmetry Breaking (RSB), overlap distribution, free energy, partition function, and the fundamental models used in spin glass theory.

[2025/04/21, 17:00] Lecture 2 slide

Abstract: This second lecture explores two facets of modern spin-glass theory. First, we examine the ultrametric organisation of Gibbs measures and show how Panchenko’s result can be encoded through the Dovbysh–Sudakov and Aldous–Hoover representation theorems. Second, we turn to spherical spin glasses and explain how the multidimensional Kac–Rice formula, combined with random-matrix techniques, yields precise asymptotics for the number of critical points—that is, the energy-landscape complexity—across different energy levels.

[2025/04/21, 17:00] Lecture 3 slide

Abstract: In this third talk, we learn about how critical point(complexity) and Gibbs measure configuration related to its free energy and energy landcscpae. Also, we learn about dynamics on spin glass model and see how these dynmiacs phase transitiona and static geometry that we learned before.

• Lecture 1 by Francesco Caravenna: 10/15 (Tue) 16:00

• Lecture 2 by Francesco Caravenna: 10/22 (Tue) 16:00

• Lecture 3 by Rongfeng Sun: 10/30 (Wed) 10:30  

• Lecture 4 by Rongfeng Sun: 11/06 (Wed) 10:30

• Lecture 5 by Nikos Zygouras: 11/14 (Thu) 17:00

• Lecture 6 by Nikos Zygouras: 11/22 (Fri) 17:00

[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.

• Lecture 1: 09/19, Thu 10:00, 129-301 

• Lecture 2: 09/26, Thu 10:00, 129-301 

• Lecture 3: 10/01, Tue 10:00, 129-301 

• Lecture 4: 10/08, Tue 10:00, 129-301

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.

• Lecture 1: 04/11, Thu 11:00, 129-104 

• Lecture 2: 04/11, Thu 14:00, 129-104 

• Lecture 3: 04/12, Fri 11:00, 129-104

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.

 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. 

• Lecture 1 (8/31 16:00, online): Introduction to the contact process

• Lecture 2 (8/31 17:00, online): Contact process on finite graphs

• Lecture 3 (9/4 16:00, online): Metastability of the contact process on lattice boxes and more general graphs

• Lecture 4 (9/5 16:00, online): Behavior on power law random graphs, metastable densities

• Lecture 5 (9/18 16:00, online): Some results on regular trees, truncated trees, random d-regular graphs and dynamic graphs

The contact process is a class of interacting particle systems modelling the spread of an infection in a finite population. In the last two decades, there has been tremendous advance in the study of this process on graphs that capture aspects of real-world populations. The goal of this mini-course is to review some highlights of the literature on the contact process, with a focus on the metastability phenomenon, which occurs when the process stays active for a very long, but ultimately finite, amount of time in a finite graph. We will start by covering the behavior of the contact process on the infinite Euclidean lattice, and well-known properties of the contact process on infinite and finite graphs. We will then specialize to finite graphs, starting from the finite-volume phase transition on lattice boxes, to illustrate the key methods and ideas. We will then discuss general metastability results that are applicable to large classes of graphs, and go over a few applications, such as the behavior of the contact process on power law random graphs. In the last part of the course we will discuss recent results on the contact process on a class of dynamic graphs (the random d-regular graph with switching edges).

• Lecture 1 (7/19, Wed 16:00, online): Metastable dynamics of one-dimensional Allen-Cahn type equations

The Allen-Cahn equation describes phase separation in an alloy, or more generally in a mixture of substances with different physical properties. It can be viewed as the continuum limit of a system of interacting stochastic differential equations, when the number of variables goes to infinity. After introducing the discrete model and recalling some known facts of its metastable dynamics, based on the so-called potential-theoretic approach, we will look at sharp asymptotics for expected transition times, of Eyring-Kramers type, for a family of parabolic stochastic PDEs, including the Allen-Cahn equation. It turns out that the prefactor of the expected transition time involves a Fredholm determinant. 

Based on joint work with Barbara Gentz,

https://dx.doi.org/10.1214/EJP.v18-1802

https://dx.doi.org/10.4171/NEWS/117/3 

• Lecture 2 (7/19, Wed  17:00, online): Renormalisation and metastable dynamics of the two-dimensional Allen-Cahn equation

Unlike its one-dimensional analogue, the two-dimensional Allen-Cahn stochastic PDE is mathematically ill-posed, and needs a renormalisation procedure to make sense. We will look at the effect of this renormalisation on sharp asymptotics of metastable transition times, of Eyring-Kramers type, for this equation. It turns out that renormalisation has the effect of replacing the Fredholm determinant by a so-called Carleman-Fredholm determinant. Based on joint work with Giacomo Di Gesù and Hendrik Weber, https://dx.doi.org/10.1214/17-EJP60 https://ems.press/books/elm/232

[2022/10/26, 15:00] Lecture 1: Introduction to statistical learning theory, neural networks, and SGD

We will cover basic concepts from statistical learning theory. More precisely, we will first formally define what statistical learning means, and we will define several notions that will be used in the later lectures. We will then introduce fully-connected neural networks and define the "empirical risk minimization" (ERM) problem within this context. To attack the ERM problem, we will introduce the optimization algorithm, stochastic gradient descent (SGD), which is one of the most popular algorithm choices for neural networks. We will finally discuss the current scientific context and identify two research questions that are introduced by SGD applied on neural networks.

[2022/11/02, 15:00] Lecture 2: The "wide minima phenomenon", heavy-tailed behavior of SGD, first exit times

We will address the first research question introduced in the first lecture. Namely, we will consider the "wide minima phenomenon", which claims that SGD automatically finds "wide valleys". We will then ask the question "why would SGD prefer wider valleys?" and formalize this question as a first-exit-time problem associated to a stochastic differential equation (SDE). After introducing the required mathematical notions for heavy-tailed SDEs, we will go over two important results from statistical physics and discuss the crucialness of "heavy-tailed behaviors" on the first-exit-times. We will finally go over several experimental results, which empirically analyze the heavy-tailed structure in SGD. This lecture is based on the article arXiv:1901.06053. 

[2022/11/09, 15:00] Lecture 3: Generalization bounds for heavy-tailed SGD

We will address the second research question introduced in the first lecture. In particular, we will aim at deriving a rigorous "generalization bound" for SGD under the assumption that it can be well-modeled by a heavy-tailed SDE. Our goal will be to formally investigate the influence of heavy-tails on the generalization performance. To do so, we will first introduce basic concepts from fractal geometry and geometric measure theory. Then, we will illustrate the advantages of heavy-tails in terms of generalization performance by proving a generalization bound. This lecture is based on the article arXiv:2006.09313. 

[2022/11/23, 15:00] Lecture 4: Negative aspects of heavy tails: mode shifts and debiasing

In Lectures 2 and 3, we illustrate the positive aspects of heavy tails in SGD, mainly in terms of the generalization error. In this lecture, we will investigate the effects of heavy tails on the training error (i.e. the empirical risk). Contrary to the previous lectures, instead of vanilla SGD, we will consider a popular variant of SGD, called SGD-momentum. We will investigate the influence of heavy-tails on training in terms of the modes of the invariant measure associated to a heavy-tailed SDE that aims to model SGD-momentum. We will observe that heavy tails can introduce an undesired bias on the training error and we will derive an alternative method to attenuate this bias. This lecture is based on the article arXiv:2002.05685. 

• Lecture 1, 12/07, Tue 10:00, online

• Lecture 2, 12/08, Wed 10:00, online

• Lecture 3, 12/09, Thu 10:00, online

The theory of hydrodynamic limits of interacting particle systems is a well developed subject in statistical mechanics. These hydrodynamic limits aim to derive solutions of PDEs as scaling limits of observables of stochastic systems such as density, energy, momentum, etc.. One can understand these scaling limits as law of large numbers on a functional space. Therefore, the corresponding CLT appears as a natural question. Surprisingly, a general approach to derive these CLTs is only available in equilibrium, namely in situations in which the underlying system is in a stationary state. The aim of this minicourse is to describe a novel approach that allows to derive such CLTs for a large class of systems and initial states. The corresponding limit turns out to be a stochastic heat equation with space-and-time-inhomogeneous noise. In order to reduce technical problems to a minimum, we will discuss in detail one particular example, the so-called reaction-diffusion model, starting from translation-invariant initial states.

• Lecture 1. (Tony Lelièvre) Introduction to the quasi-stationary distribution approach for the exit problem: theoretical and numerical aspects

- Part 1:  Aug 25th (Wed), 09:00-11:00 in France and 16:00-18:00 in Korea

- Part 2:  Aug 26th (Thu), 08:00-09:00 in France and 15:00-16:00 in Korea

• Lecture 2. (Dorian Le Peutrec) Semi-classical analysis and Witten Laplacians

- Part 1:  Aug 26th (Thu), 09:00-11:00 in France and 16:00-18:00 in Korea

- Part 2:  Sep 1st (Wed), 09:00-10:00 in France and 16:00-17:00 in Korea 

• Lecture 3. (Boris Nectoux) Analysis of the exit point distribution using a semi-classical approach

- Aug 27th (Fri), 09:00-11:00 in France and 16:00-18:00 in Korea 

See also https://sites.google.com/view/qsd2021.

We consider the exit event from a metastable state for a stochastic process. We show how, using the concept of quasi-stationary distribution, one can model the exit event from a metastable state by a jump Markov model. For the overdamped Langevin dynamics and in the limit of small noise, we show that the jump Markov model can be parameterized by the Eyring-Kramers formulas. This mathematical analysis is useful from a modeling and a numerical viewpoint. Indeed, it justifies the use of jump Markov models (kinetic Monte Carlo or Markov State Models) with jump rates determined using the Eyring-Kramers formula (Harmonic Transition State Theory) to describe the evolution of a molecular system over long timescales. This study is also motivated by the design and analysis in terms of accuracy and efficiency of so-called accelerated dynamics algorithms (developed in particular by D. Perez, A.F. Voter and collaborators at Los Alamos National Laboratory), which use the approximation by a jump Markov model to simulate metastable trajectories of the Langevin or overdamped Langevin dynamics over large timescales. The lectures will be divided into three parts. The first lecture will be devoted to an introduction to the quasi-stationary distribution approach for the exit problem. We will present the main motivation for this approach, the main result on the exit point distribution for the overdamped Langevin, as well as numerical counterparts. The second lecture will provide some essential tools from the semi-classical analysis of Witten Laplacians. We will discuss in particular some results on the spectrum of the infinitesimal generator of the overdamped Langevin dynamics in the small noise limit. Finally, the third lecture will give a detailed description of the first exit point distribution of the overdamped Langevin dynamics in the small noise regime, using in particular the techniques introduced in the second lecture.

References

1. H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrödinger Operators, Springer, 1986.

2. M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, Cambridge University Press, 1999.

3. G. Di Gesù, T. Lelièvre, D. Le Peutrec and B. Nectoux, Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach, Faraday Discussion, 195, 469-495, 2016.

4. G. Di Gesù, T. Lelièvre, D. Le Peutrec and B. Nectoux, Sharp asymptotics of the first exit point density, Annals of PDE, 5(1), 2019.

5. B. Helffer, M. Klein and F. Nier, Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp. 26, 41-85, 2004.

• Lecture 1, 4/19, Mon 13:30, online

Random matrix theory has been greatly successful in last a few decades with remarkable results. One of the most important results in the random matrix theory is universality, which asserts that the global/local statistics of random matrix eigenvalues is universal. In this talk, I will give an overview of the random matrix theory and introduce the concept of the universality. Several key theorems and main ideas will also be discussed.

• Lecture 2, 4/26, Mon 13:30, online

Random matrices with Gaussian random variables are the most basic and fundamental ones in random matrix theory. In this talk, I will explain the key results on Gaussian unitary ensemble (GUE) and Gaussian orthogonal ensemble (GOE) and introduce the method based on orthogonal polynomials for the analysis of random matrices.

• Lecture 3, 5/03, Mon 13:30, online

Gaussian unitary ensemble (GUE) and Gaussian orthogonal ensemble are naturally extended to Wigner matrices. In this talk, I will explain the main changes for the key results on Wigner matrices by comparing them with those on Gaussian ensembles. I will also introduce the method based on the Stieltjes transform for the analysis of random matrices.

• Lecture 4, 5/10, Mon 13:30, online

Most of the results in random matrix theory were proved for the simplest models, typically the Gaussian ensembles, and extended by comparing the given model with the known models. In this talk, I will introduce various comparison methods including the Lindeberg replacement trick, the Dyson matrix flow, and interpolation.

• Lecture 5, 5/17, Mon 13:30, online

Most of the results in random matrix theory were proved for the simplest models, typically the Gaussian ensembles, and extended by compaMany results and techniques for Wigner matrices are useful in other random matrix models, which can be regarded as the universality in a higher level. In this talk, I will introduce various random matrix models appearing in various fields and explain several important results for them. Several interesting applications and open problems will also be discussed.ring the given model with the known models. In this talk, I will introduce various comparison methods including the Lindeberg replacement trick, the Dyson matrix flow, and interpolation.