Notes
Collection of digitised notes since 2017,
listed in reverse chronological order.
Highlighted Notes
English, LaTeX
An exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold, with an account of the physical background and the deformation theory via differential graded Lie algebras and L_infinity-algebras.
English, LaTeX
These are summary notes on the 3rd-year functional analysis course at Oxford. The main theory is devoted to studying the bounded linear operators between normed vector spaces. Some extra material on the Hahn-Banach theorem and dual spaces are included for the sake of self-containedness.
Some notes on the 3rd-year course representation theory at Oxford. The main topics are the structure of semi-simple modules, character theory, and the induced representations.
Chinese, LaTeX
English, LaTeX, in progress
These notes intend to cover the mainline of undergraduate algebra courses, including the fundamentals of commutative rings, fields, and modules.
Chapter 1 and 2 on basic ring theory, and a portion of Chapter 3 on field extensions have been fully completed. The rest of Chapter 3, Chapter 4 on modules, and Chapter 6 on commutative algebra have been completed except for proofs. Chapter 5 on linear algebra is still empty.
English, LaTeX
These notes mainly follow the 2nd-year course complex analysis at Oxford. It includes the basic materials, such as Cauchy integrations, Laurent series and isolated singularities, and residue calculus, and some off-syllabus topics such as the Weierstrass Factorisation Theorem and the proof of the Riemann Mapping Theorem.
Chinese, Scanned
Hand-written 200 pages notes on Linear Algebra, which took me more than a year to complete. As an extended study before entering college, the notes contain material ranging from solving linear equations by augmented matrices to the universal property of tensor products.
Linear Algebra was my first mathematical subject. I started to read Sheldon Axler's Linear Algebra Done Right in Grade 11, the beautiful and artistic style of which lead me to the world of pure mathematics.
Other Notes
Mini-Project for the MMathPhys course String Theory I. A brief introduction to the path integral and BRST quantisation of bosonic strings.
English, LaTeX
A brief notes on my summer research project in 2021, as a refinement of the work by V. V. Geogjaev (1984). We studied the dispersionless limit of the KdV equation as an example of the inverse scattering problem of the Schrödinger equation in the WKB approximation.
English, LaTeX, in progress
Notes that intend to cover the basic category theory.
English, LaTeX, in progress
English, LaTeX
Rewritten lecture notes on the 3rd-year course algebraic curves at Oxford. They closely follow the book Complex Algebraic Curves by Frances Kirwan.
Some brief revision notes on the 2nd-year courses Differential Equations II, Integral Transforms, and Calculus of Variations. Proofs and calculation details are omitted.
English, LaTeX
A part of the notes on the 2nd-year course Further Quantum Mechanics at Oxford. The main topics are approximation methods and atomic physics.
English, Scanned
Chinese, Scanned
A brief and incomplete notes on the foundation of analysis. The topics include naïve set theory, real numbers, sequences and limits, and continuous functions.
The notes were made in Grade 12 and my fresher year before transferring from physics to mathematics.