Lecture notes on the Gaussian Free Field

Lecture notes on the DiPerna-Lions theory in abstract measure spaces

These notes closely correspond to a series of lectures given by the first author in Toulouse, on the recent extension of the theory of ODE well-posedness to abstract spaces, jointly obtained by the two authors. In the last part, we describe some further developments with respect to the theory of (possibly degenerate) diffusion processes, in a similar setting, contained in the second author's PhD thesis.

Lecture Notes on Gradient Flows and Optimal Transport

We present a short overview on the strongest variational formulation for gradient flows of geodesically $λ$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.

Lecture Notes on Stochastic Processes

This is lecture notes on the course "Stochastic Processes". In this format, the course was taught in the spring semesters 2017 and 2018 for third-year bachelor students of the Department of Control and Applied Mathematics, School of Applied Mathematics and Informatics at Moscow Institute of Physics and Technology. The base of this course was formed and taught for decades by professors from the Department of Mathematical Foundations of Control A.A. Natan, S.A. Guz, and O.G. Gorbachev. Besides standard chapters of stochastic processes theory (correlation theory, Markov processes) in this book (a…

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…

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 "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。