Lecture Notes on Random Matrix Theory

Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity

The purpose of these notes, based on a course given by the second author at Les Houches summer school, is to explain the probabilistic construction of Polyakov's Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, these notes contain a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. These notes are rather short and require no prior knowledge on the topic.

Lecture notes: Efficient approximation of kernel functions

These lecture notes endeavour to collect in one place the mathematical background required to understand the properties of kernels in general and the Random Fourier Features approximation of Rahimi and Recht (NIPS 2007) in particular. We briefly motivate the use of kernels in Machine Learning with the example of the support vector machine. We discuss positive definite and conditionally negative definite kernels in some detail. After a brief discussion of Hilbert spaces, including the Reproducing Kernel Hilbert Space construction, we present Mercer's theorem. We discuss the Random Fourier Featu…

量子扩散与随机矩阵理论讲义

Lecture notes on Quantum Diffusion and Random Matrix Theory

本讲义介绍一种基于随机矩阵理论研究随机薛定谔方程扩散极限的新方法，内容源自作者与 Adam Black 和 Reuben Drogin 的合作工作，并在索邦大学暑期学校讲授。

Lecture Notes on "Random Matrices"

This in an introduction to random matrix theory, giving an impression of some of the most important aspects of this modern subject. In particular, it covers the basic combinatorial and analytic theory around Wigner's semicircle law, featuring also concentration phenomena, and the Tracy-Widom distribution of the largest eigenvalue. The circular law and a discussion of Voiculescu's multivariate extension of the semicircle law, as an appetizer for free probability theory, also make an appearance. The material here was presented in the winter term 2019/20 at Saarland University in 24 lectures of 9…

《自由概率论》讲义

Lecture Notes on "Free Probability Theory"

本文介绍《自由概率论》讲义内容，涵盖基本理论及与随机矩阵、算子代数的关系，基于授课者两本著作，限定标量值框架，2018/19冬季学期在萨尔兰大学授课并录制上线。

关于粗糙路径及其在机器学习中应用的讲义

Lecture notes on rough paths and applications to machine learning

本讲义系统介绍 signature 与粗糙路径理论的基础，并综述其在核方法、神经粗糙微分方程等机器学习方向的应用。

Lecture notes on the Gaussian Free Field

The Gaussian Free Field (GFF) in the continuum appears to be the natural generalisation of Brownian motion, when one replaces time by a multidimensional continuous parameter. The goal of these lecture notes is to describe some aspects of the continuum GFF and of its discrete counterpart defined on lattices, with the aim of providing a gentle self-contained introduction to some recent developments on this topic, such as the relation between the continuum GFF, Brownian loop-soups and the Conformal Loop Ensembles CLE(4). This is an updated and expanded version of the notes written by the first au…