Sung-Soo Byun (SNU)

Title: Integrable Random Matrices through Differential Equations

Abstract: In this lecture series, I will introduce how the techniques using differential equations can be applied to derive statistical properties of eigenvalues of random matrices.

Lecture 1. Spectral Moments: Harer-Zagier Formulas

In the first lecture, I will present a methodology for examining the spectral moments of random matrices. This approach hinges on an understanding of the recurrence relation governing these spectral moments, known as the Harer-Zagier formula. Notably, these formulas stand as a prominent illustration, showcasing the inherent combinatorial and topological significance embedded in the random matrix statistics.

Lecture 2. Eigenvalue Correlations: Generalised Christoffel-Darboux Identities

In the second lecture, I will introduce a way to study eigenvalue correlations of random matrices using the generalised Christoffel-Darboux formula. The classical Christoffel-Darboux formula is an identity for a sequence of orthogonal polynomials with respect to the weighted Lebesgue measure on the real axis. It plays a crucial role in studying the correlation functions arising in the context of unitarily invariant Hermitian random matrix theory. I will present the generalised Christoffel-Darboux formula, allowing the asymptotic analysis of some non-Hermitian random matrices. Taking the elliptic random matrix as our main example, I will present several implementations of this formula.

Lecture 3. Universality: Ward’s Equations

Ward’s equations in conformal field theory describe the perturbation of correlation functions under the insertion of a stress tensor. Its discrete analogue for the finite Boltzmann-Gibbs measure is often known as the loop equations in random matrix theory. In the third lecture, I will introduce how the theory of Ward’s equations can be implemented to study the local universality conjecture of random matrices.

Daesung Kim (Georgia Institute of Technology)

Title: Stability for the logarithmic Sobolev inequality and isoperimetric type inequalities from the stochastic analysis

Abstract: Stability for functional and geometric inequalities has been of great interest to many researchers in analysis and probability. For an inequality whose optimizers are fully understood, we are interested in the deviation from optimizers when the equality is close to being achieved. There has been considerable progress in this direction: the Sobolev inequalities, the Hardy--Littlewood--Sobolev inequality, the logarithmic Sobolev inequality, the Hausdorff--Young inequality, the isoperimetric inequalities, and the Faber--Krahn inequality.  In this lecture series, I will focus on the stability results of the logarithmic Sobolev inequality and spectral inequalities arising from the stochastic analysis. 

Lecture 1: Stability of the log Sobolev inequality

The logarithmic Sobolev inequality, which states that the Fisher information is bounded below by the relative entropy, has been extensively studied in analysis and probability.  In the Euclidean space, equality holds if and only if a measure is Gaussian.  After the equality case is fully characterized by E. Carlen, we are interested in measuring how far a measure is away from Gaussian measures when it is close to achieving the equality.  In this talk, we discuss a recent progress on the stability for the logarithmic Sobolev inequality. 

Lecture 2: Quantitative isoperimetric type inequalities from the stochastic analysis

Saint-Venant inequality says that the torsional rigidity of a region is maximized when the region is a ball with the same volume. The torsional rigidity can be understood as the expected life time of the Brownian motion starting at uniform distribution in a region. It turned out that such inequality holds for a general class of quantities from stochastic analysis. In particular, it was shown by Banuelos and Mendez-Hernandez that a Schrodinger semigroup associated with a general class of Levy processes are maximized when the function and the process are symmetrized. As a consequence, one can see that some quantities related to Levy processes, such as the torsional rigidity, the expected lifetime, and the heat content are maximized when the region is a ball. For most of these inequalities, it is still open to characterize the equality cases and obtain quantitative improvements. In this talk, we discuss some progress along this line and interesting open problems. 

Kunwoo Kim (POSTECH) 

Title: Introduction to stochastic heat equations and its properties 

Talk 1:  Introduction to stochastic heat equations (SHEs)
Abstract: Stochastic heat equations (SHEs) usually refer to heat equations perturbed by noise, and the noise represents random external force or random potential. In this talk, we investigate existence, uniqueness, and regularity of a solution to SHEs. 

Talk 2: Comparison principles, positivity, and compact support property for SHEs. 
Abstract: This second talk focuses on various properties of SHEs. First, we consider some comparison principles such as the pathwise and moment comparison. The pathwise comparison principle guarantees non-negativity of solutions to SHEs when the noise coefficient hits 0. We then examine conditions under which the solution is strictly positive everywhere, or alternatively, it has a compact support for all time.

Talk 3: Small ball probabilities and Chung's law for SHEs 
Abstract: Small ball probabilities refer to the probabilities that the solution is within a small ball under certain norms. In this talk, we consider one-dimensional SHEs on a torus and investigate a recent progress on small ball probability estimates for SHEs. We also identify small ball constants for certain cases and investigate the Chung type laws of the iterated logarithm for SHEs. 

Kyeongsik Nam (KAIST)

Title: Universality of log-correlated fields

1. Maximum of log-correlated fields
I will talk about truncated second moment method to study the maximum of log-correlated fields. It requires the Ballot theorem for random walks and sprinkling techniques.

2. Local structure of log-correlated fields
I introduce Liggett's characterization of the Poisson point process and how it can be utilized to analyze the local energy landscape of log-correlated fields.

3. Aspects of non-Gaussian log-correlated fields
I will talk about the recent progress on some non-Gaussian log-correlated models, such as random matrices and Ginzberg-Landau models.

Yilin Wang (IHES) 

Title: Brownian loop measure and determinants of Laplacians

Abstract: Many concepts in complex analysis can be translated into probabilistic objects in the plane because the Laplacian is the generator of Brownian motion. I will focus on the link between the Brownian loop measure and determinants of Laplacians in the plane and show a few applications using this link.

Lecture 1. Connecting Brownian loop measure and determinants of Laplacian

We explain how the Brownian loop measure can be related to the log determinants of Laplacian. This is based on the works of Le Jan, Dubédat, and Lawler-Werner.

Lecture 2. Applications

The Loewner energy can be expressed using determinants of Laplacian and therefore has an interpretation in terms of Brownian loop measure. This explains why the Brownian loop measure appears in the variational formula of the Loewner energy. On the other hand, the Brownian loop measure is central to the conformal restriction property of SLE which also gives one way to interpret the "central charge'' of SLE. From this, we will show how we can derive the Onsager-Machlup functional of SLE loop measure and representations of Virasoro algebra using SLE loop measure.  This is based on my work, joint work with Sung, with Cafagnini, and an ongoing work with Gordina and Qian.