# Teaching

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Teaching at NUS

2023 Spring: MA4221-Partial Differential Equations (Undergraduate).

2024 Spring: MA4221-Partial Differential Equations (Undergraduate).

2025 Spring: MA5213-Advanced Partial Differential Equations (Graduate).

#### I received the "Professor Joel Dean Excellence in Teaching" award for TAs in 2021.

2017 Fall: Method in Complex Analysis, Calculus 1 (Eng).

2018 Spring: Partial Differential Equations.

2018 Fall: Calculus 2 (Eng).

2019 Spring: Honour Analysis 2, Calculus 2 (Eng).

2019 Fall：Honour Analysis 1, Real Variables.

2020 Spring: Honour Analysis 2, Partial Differential Equations.

2020 Fall: Honour Analysis 1, Differential Equations and Applications.

2021 Fall: Real Variables, Introduction to Proofs.

2022 Spring: Honour Analysis 2, Differential Equations and Applications.

#### I received the "Outstanding TA in USTC" award in 2016(rank 6/703, score 4.95/5.00) and 2017(rank 7/562, score 4.98/5.00).

2017 Spring: Differential Equations II, Instructor: Professor Xing Liang.

2016 Fall: Advanced Real Analysis, Instructor: Professor Hao Yin.

2016 Spring: Real Analysis (H), Instructor: Professor Hao Yin.

- My Teaching Preferrences: PDE(Grad)>Harmonic Analysis>Real Analysis(Grad)>PDE(Undergrad)>Real Analysis(Undergrad)>ODE>Mathematical Analysis>Functional Analysis>>Others. Here are some reference books for undergrad and junior grad courses on analysis and PDEs, based on the training plan for undergraduate students at Univ. of Sci. & Tech. of China (USTC).

- At USTC, fall semester is usually 16~18-week and spring semester is usually 15~16-week.
- Mathematical analysis (A1)-(A3) (compulsory, Sem1 Year1~Sem1 Year2). It is a proof-based, 3-semester course. It basically covers single-variable calculus (A1), multi-variable calculus (A2), number series and function series (including Fourier series), improper integrals (A3). Textbook:《数学分析教程（第三版）》常庚哲、史济怀，中国科学技术大学出版社。A good addendum to Fourier theory is Stein's book "Fourier Analysis".
- Differential equation I (compulsory, Sem1 Year2). It covers basic ODE theory (8 weeks) and classical PDE theory (8 weeks). Textbooks: ODE (柳彬《常微分方程》or 丁同仁、李承治《常微分方程教程》), PDE (周蜀林《偏微分方程》, Evans PDE Chatper 2, 4.)basically covers:
- solvability of basic types of ODEs, implicit ODEs, singular solution and envelops;
- existence theorem (Picard iteration, continuous dependence on given data);
- high-order linear ODEs and 1st-order ODE systems;
- an introduction to planar dynamical systems (stability of equilibria, Lyapunoiv method, Poincare-Bendixon theorem*, structual stability and bifurcation*);
- solvability of transport equations and multi-dimensional linear wave equations, finite propagation speed;
- heat equation: Fourier method, maximum principle;
- separation of variables;
- Laplace equation: properties of harmonic functions (mean value, maximum principle, Harnack's inequality, gradient estimates), Poisson equations (R^d, Green's function method).

- Real Analysis (compulsory, Sem2 Year2). Textbook: Stein Ch 1-3, 6 (Lebesgue measure, Lebesgue integrals, differentiation theory, abstract measure) plus more about convergence theorems and Lp spaces.
- Complex analysis (compulsory, Sem2 Year2). Textbook: 《复变函数》史济怀，中国科学技术大学出版社, basically covers Cauchy's integral formula, complex series and Laurent series, maximum principle and Schwarz lemma, infinite products, holomorphic extension, Riemann mapping theorem.
- Functional Analysis (compulsory for pure math major and prob & stat major, optional for applied math and computational math, Sem1 Year3). Textbook: Chapter 1,2,4 of 《泛函分析教程（上册）》张恭庆，北京大学出版社. (Banach/Hilbert Space, Hahn-Banach thm and related, weak convergence, spectrum of compact operators) My personal preference:Ch1-6 in "Functional analysis", Vol. 1 of "Methods of Modern Mathematical Physics" by M. Reed and B. Simon.
- Advanced Real Analysis (optional, recommended Sem1 Year3 or above). Textbook: Folland Real analysis Chapter 1-3, 6-9. (abstract measure, signed measure, Riesz representation theorem, Lp space and interpolation, Fourier transform, generalized functions and Sobolev spaces.)
- Differential equation II (optional, recommended Sem2 Year3 or above). Major part: Evans Part II. Optional part (depends on who is the instructor): part of Evans Ch 8 and 9, materials concerning wave equations (Sogge's book, Jonathan Luk's notes, Qian Wang's notes) and dispersive equations (Strichartz estimates for NLS), material concerning elliptic PDEs (techniques of gradient estimates and construction of auxiliary functions).
- Harmonic Analysis (optional, recommended Sem2 Year3 or above). Major references are: Stein's singular integral book, harmonic analysis book; Vol. 1 of Muscalu-Schlag; Grafakos' books (GTM 249, 250).
- Basic tools: Hardy-littlewood maximal function, Fourier transform. (cf. Duoandikoetxea)
- Calderon-Zygmund Singular Integral (cf. Stein's singular integral book Ch 1-4)
- Littlewood-Paley and almost orthogonality (cf. Muscalu-Schlag, Ch 8-9)
- H1 and BMO (cf. Duoandikoetxea)
- Oscillatory Integrals (cf. Stein's"bible" Ch 9-10, Muscalu-Schlag Ch 11).

- 2nd-order linear elliptic PDEs (optional, recommended Sem1 Year3 or above). Textbooks: Elliptic PDE by Gilbarg and Trudinger (Ch 1-9) or Elliptic PDE by Qing Han and Fanghua Lin.
- Several-variable complex analysis (optional, recommended Sem1 Year3 or above).
- various topic courses, not regularly offered (e.g., nonlinear elliptic PDE, optimal transport, nonlinear dispersive or fluid PDEs, differential dynamical systems...)

The following courses related to analysis and PDEs are regularly offered at USTC: