Note for Riemannian Surface
This is the note on Riemannian surface. The reference textbook is “Riemannian Surface” by Mei Jiaqiang. The note is divided into two parts.
Part I: Basic definition of Riemann surfaces, uniformization theorem, Riemann-Roch theorem.
Part II: Cohomology and complex geometry of surfaces.
Probability Theory Lecture Note
This note is based on the course “Probability Theory(MATH3007)” of USTC, taught by Prof. Dang-Zheng Liu. The note covers all the content in the course, including the basic definitions in the probability theory, discussion on random variables, law of big numbers and central limit theorem.
Note for Morse Theory
This is the note on Morse Theory. The reference textbook is “Morse Theory” by J. Milnor.
Note for Functional Analysis(H) (2024FALL)
Note for the FA course. Coverage: Chap 1-3 of the textbook by Zhang Gongqing
Note for Differential Geometry(H) (2024FALL)
Note for the DG course. Coverage: Chap 1-5,7,8.1-8.3 of the textbook by Chen Qing.
Note for Complex Geometry(NOT COMPLETED)
This is the note that I made when learning the video course of complex geometry taught by Prof. Zhang Xi in 2020SP semester. The video can be found here. The note is divided into two parts.
Part I: Introduction to multivariable complex analysis, basic definition of complex manifold and Kahler manifold, connection and curvature, differential operator on complex manifold.
Part II: Hodge theory, Chern-Weil theory, sheaf theory and stability of vector bundle, line bundle and vanishing theorem, Hermitian-Einstein bundle, Donaldson-Uhlenbeck-Yau theorem and Calabi-Yau theorem.
One may consult “Differential Geometry of Complex Vector Bundles” by Kobayashi for Chapter 1 to 7 of this course.
Note for the Seminar in Characteristic Class Theory(Ver. 20240728)
This note is in characteristic class theory for the seminar that is expected to be held next semester. It is still very incompleted and under working. I will upload the newest version here in time.
Note for Algebraic Geometry I(MATH5016P.01 2024FALL)
Note for the AG1 course.
Chap 1: Variety(basic definitions, Hilbert Nullstellensatz, polynomial map and coordinate ring, normal function, function field, rational/birational/dominant map, dimension and smoothness)
Chap 2: Sheaf and scheme(basic definitions, Cartier divisor and invertible sheaf, getting used to the language and notation)
Chap 3: Intersection theory on projective plane(Artin ring, Max-Noether theorem, intersection number and its calculation, Bezout theorem)
Chap 4: Normalization and resolution of singularities
Chap 5: Riemann-Roch theorem(Weyl divisor on curve and its connection with Cartier divisor, linear system, differential and canoncial divisor, proof of R-R theorem)