Below are some of the courses I have taken, along with my reflections on the material:
https://dec41.user.srcf.net/notes/ These are the famous Cambridge Notes. They contain an extensive collection of notes on mathematics courses. I particularly recommend the sections on optimization and stochastic calculus. However, it is worth noting that some of the material is not suitable for beginners. Please use them with discretion.
https://github.com/Triang-jyed-driung/Analyse-Mathematique-Yu-Pin-Analysis-123/ This is the textbook used by the Tsinghua University. One interesting textbook for a beginner to read , and it also contains some interesting viewpoint to other anlysis-associated questions. Note this textbook is in Chinese.
https://link.springer.com/book/10.1007/978-3-032-17635-6 This is a textbook for learning and review the Tensor Network Method. Not recommended for auidence with no background.
https://www.cambridge.org/core/books/tensor-decompositions-for-data-science/640814D308696CD61CB9112EA57B2911 This is the textbook introducing tensor network methods as applied to statistics. I've always thought that multilinear algebra methods, such as Tensor Train, are a natural way of representing high-dimensional data — even though, theoretically, they are somewhat limited to Hilbert space. Their advantage lies in capturing local relationships between coordinates, which is why physicists first applied them to one-dimensional quantum systems (e.g., DMRG). A major disadvantage of multilinear algebra methods, however, is that they struggle to represent long-range relationships. One can show that even in the simplest cases involving long-range dependencies, Tensor-Train-type methods require the rank to grow exponentially in order to capture them.
Analysis
Real Analysis
Complex Analysis
Fourier Analysis
Functional Analysis
Optimal Transport
Tensor Network Method
Numerical Analysis
Dynamical System
Numerical Analysis
Numerical Methods for DE
Numerical Methods for PDE
Uncertain Quantification
Lattice Boltzman Method (Audit)
Optimization
Operation Research
Nonconvex Global Optimization
Foudations of Opt for Machine Learning (Lecturer: Prof. Suvrit Sra)
Opt for Machine Learning (Lecturer: Prof. Martin Jaggi)
Combinatorial Optimization (Audit)
Convex Optimization (Textbook from Boyd)
Probablity & Statistics
Probablity Theory
Branching Random Walk
Molecular Dynamics Simulation and Deep Learning
Statistics
Economic Statistics
Foundamentals of Mathematical Statistics
Deep Learning Theory (Lectuer: Prof. Zhiyuan Li)
Statistical Foundation for Machine Learning (Audit)
Computer Science
Foundations of Data Analysis
Foundations of Generative AI
Natural Language Processing
Data Structure & Algorithm
Computational Single-Cell Method
(And all the theoritical CS courses mentioned above)
Physics
Mechanical Mechanics
Electromagnetism
Waves and Modern Physics
Quantum Mechnics
Particle Physics I (Audit)
Others
Algebra
Topology