This piece was written in the second semester of my third year, in response to a survey by Professor Yan Kai soliciting opinions on how the school trains its students. Professor Yan wanted us to speak our minds plainly. Some of the evaluations may be somewhat intemperate, but they more or less reflect my state of mind and understanding at the time. The evaluations are for reference only.

The first three semesters

Mathematical Analysis

Teacher: Tang Yanbin (first semester), Huang Yongzhong (second and third semesters), Zeng Haozhi (problem sessions) Textbook: Cui Shangbin. Time: from first year through the first semester of second year, three semesters in total Evaluation: Both teachers were very conscientious. Professor Huang put a lot of care into it and would hand out plenty of supplementary material. I used to fire away at Cui Shangbin’s textbook and told every junior I met to switch to another book, but now I can genuinely appreciate Professor Huang’s good intentions — a textbook with no answers really does matter. At the beginning, the theory of the real numbers was hard to accept, and I think the main issue was the logical starting point: we had not yet realised where the logical starting point lay. We could accept that one had to begin with the real numbers, but it was not clear why it had to be made so complicated. It was all a muddle. As I went deeper into mathematics and my logical training deepened, though, I stopped worrying about it — it really does have to be done this way. One complaint, though: the problem sessions were not very effective. As a beginner there was a great deal I was muddled about, and once the teacher went a little deeper I simply could not follow. It would have been better to concentrate on going through the homework. In the second semester of first year, when classes were online, the teacher would post the homework solutions, and I actually think that was when the problem sessions helped most.

Advanced Algebra and Analytic Geometry

Teacher: Liu Xianzhong Textbook: Wang Efang. Time: the whole of first year Evaluation: My study of advanced algebra, one could say, went on for a very long time — I must have gone through it at least four times. The first pass was following the teacher, in a complete muddle. The second was working through MIT’s and Gilbert Strang’s online courses and notes along with David C. Lay’s Linear Algebra and Its Applications, which gave me a great deal of intuition — especially about the meaning of matrix multiplication, the meaning of similarity, what diagonalisation does, and the various matrix factorisations — though that pass was only a cursory read. The third was reading Axler’s Linear Algebra Done Right, which gave me some grasp of the core of linear algebra, linear spaces and linear transformations; at that point I could hold forth about linear algebra, so to speak, going “from thick to thin”: nothing but linear spaces and linear maps, and the equivalence relations and canonical forms of the various linear maps. The fourth was after studying functional analysis and reading Li Jiongsheng’s Linear Algebra, when it became “thicker and thinner again”. In any case it is always rewarding to revisit. How linear algebra should actually be taught is a genuinely big question, but if you come out of the first pass completely muddled, that is worth reflecting on.

Ordinary Differential Equations

Teacher: Liu Bin Credits: 4.5. Hours: 72. Time: first semester of second year Textbook: “no textbook, only reference books” Evaluation: Professor Liu actually lectured very well; it was my own fault for not attending properly or taking notes, obsessed as I was with reading on my own — except that my own reading was only superficial and I neglected exercises, so I learned it very poorly. It was my first specialised course, unlike the foundational courses such as analysis and advanced algebra, and my way of studying really did need adjusting.

Second semester of second year

Abstract Algebra

Teachers: Wang Baowei (group theory), Chen Bo (ring theory) Credits: 4. Hours: 64 Textbooks: Xiong Quanyan; N. Jacobson, Basic Algebra Evaluation: The breadth was acceptable. Group theory covered a reasonable amount — group actions, the two isomorphism theorems and Sylow’s theorems were all covered. In ring theory we got through about four fifths of Chapter 2 of Basic Algebra. In truth there is a great deal more that could be taught in abstract algebra, but there were not enough hours — Galois theory was never even reached.

Real Analysis

Teacher: Ming Ju Credits: 4.5. Hours: 72 Textbooks: Zhou Xingwei; Zhou Minqiang Evaluation: The breadth was fine too, and everything that should have been covered more or less was — Lebesgue measure, the Lebesgue integral, the fundamental theorem of calculus and L^p spaces were all completed. The teacher’s pacing was good as well, and one thing he did very well: he had a course homepage where he posted his own lecture notes, some supplementary material, homework solutions and past exam papers. Other teachers set up QQ groups to distribute this material, which is also fine, but having a homepage does feel a little better.

Complex Analysis

Teacher: Liao Junjun Credits: 4.5. Hours: 72 Evaluation: The breadth of content was fine. Thinking back now, what did we actually cover? The core was analytic functions, Cauchy’s integral formula and its string of corollaries, the residue theorem, Laurent expansions, conformal mappings, and the properties of harmonic functions. But it felt light on the analytical side; mostly it was computation.

Probability Theory

Teacher: Wu Fuke Credits: 4.5. Hours: 72 Textbooks: Li Xianping, Foundations of Probability Theory; the teacher’s own slides Evaluation: The content is what it is, and everything that should have been taught was taught; the limit theory perhaps was not covered very deeply. Li Xianping’s book is very good — a fine textbook. The teacher lectured well too, and told plenty of stories — his own experiences, stories of mathematicians — which was very helpful. One thing though: the final exam was too easy, which meant the result ended up mattering a lot. I made simple arithmetic slips on two or three questions and ended up with only 86, even though I think I had learned the material rather well.

First semester of third year

Partial Differential Equations

Teacher: Duan Zhiwen Credits: 4. Hours: 64. Time: first semester of third year Textbook: Chen Zuchi, Partial Differential Equations Evaluation: Too little content, and not enough depth. Only the solution methods and a few properties of the three classic equations. Mainly the teacher went far too slowly — Chapter 3 (the wave equation) dragged on for nearly two months (two sessions a week). Generalised functions were never touched, the calculus of variations only barely, and I never felt any connection between PDEs and functional analysis. Is this supposed to produce mathematicians?

Functional Analysis

Teacher: Zheng Quan Credits: 4.5. Hours: 72. Time: first semester of third year Textbook: Wang Shengwang and Zheng Weixing, An Outline of Real Analysis and Functional Analysis Evaluation: The best course I have attended in three years — I cannot praise Professor Zheng Quan enough. It had both depth and breadth, the mathematical ideas were clear and lucid, and the history of mathematics and the stories of mathematicians were woven in just right (always saving the stories for the last ten minutes or so of class). (The PDE teacher also loved telling stories, but the drawback was that there were too many of them and they disrupted the schedule, and many had nothing to do with mathematics.)

Topology

Teacher: Zhang Ning Credits: 4. Hours: 64. Time: first semester of third year Textbooks: Xiong Jincheng, Lectures on Point-Set Topology; Munkres, Topology Evaluation: The depth and breadth were sufficient, and a great deal was covered — that deserves real credit. Quite a lot of both point-set topology and algebraic topology was taught; if the topology taught to applied mathematicians cannot even reach algebraic topology, what is the point of it! But the teacher’s technique was not mature enough — not like Professor Zheng Quan explaining functional analysis in simple terms. In fact many details could perfectly well have been left for the students to work through after class. All in all the teaching was not very effective, but it did push me to chew through Munkres myself, and I read a fair amount of topology. Professor Zhang Ning is a very good person who cares about his students, but his lecturing technique could do with improvement.

Mathematical Statistics

Teacher: Pan Deng Credits: 4. Hours: 64. Time: first semester of third year Textbook: Mao Shisong, Mathematical Statistics Evaluation: On the whole it was unremarkable and not especially engaging. Looking back, not much was actually taught: only point estimation, interval estimation and hypothesis testing — the first four of the book’s five chapters, about five sixths of it. I do not think this textbook handles hypothesis testing well, because it goes straight to the Neyman–Pearson standard theory: the part that derives the test is easy enough to follow, but the important notion of the p-value is very hard to grasp that way. Only after reading Chen Xiru’s A Course in Mathematical Statistics did I understand what a natural idea the p-value is. I would strongly recommend teaching hypothesis testing the way Chen Xiru does in that course: telling it in historical order — it all began in the twentieth century anyway, so the thread is fairly clear — which brings out the statistical thinking better and leaves a deeper impression. At the end one could also introduce some analysis of variance and Wald’s statistical decision theory.

Numerical Analysis

Teacher: Huang Chengming Credits: 3.5. Hours: 48+8 Evaluation: The textbook has a great many logical errors, though the teacher did mention this in class. The teaching was decent. But the same old problem: this course gives you a shallow taste of Matlab, that course a shallow taste of Python, and all you learn is copy-pasting and calling a function.

Discrete Mathematics

Teacher: Huang Aiqun Credits: 3. Hours: 48 Evaluation: A waste of time, with no depth. You could read it yourself in a day or two — what do you need 48 hours for? Is this supposed to produce mathematicians? Better to use those hours for something more closely connected to mathematics: Galois theory from abstract algebra, algebraic topology — neither of which we have hours to cover. Or swap it for mathematical logic — Professor Hu Yining could teach mathematical logic!

Update, 2023-04-19: That said, giving an introduction to graph theory is one of its few merits. Graph theory is an important tool that can describe a great many things — how can someone in a mathematics department not know it? But the standard of teaching is really too low and the content too shallow. The design of the syllabus is a very serious problem.

Second semester of third year

Introduction to Information Theory

Teacher: Liu Haixia Credits: 3. Hours: 40+8 Evaluation: I felt it was of little use, with no depth — an engineering-level treatment — and its addition to the curriculum felt rather forced: it does not echo the courses before or after it, and I did not really understand some of the background details from communications. I only came away with a smattering of information theory; at best it can serve as a small application of probability theory, but not much mathematics was taught. Covering even a few of Shannon’s theorems and the sampling theorem would have been something. In short, better to spend those hours on something more closely connected to mathematics: Galois theory from abstract algebra, algebraic topology — neither of which we have hours to cover.

Update, 2023-04-19: I now retract my complaint that it is useless and that adding it to the curriculum was too forced. Information theory is very useful, and it is likewise an important language for describing many things. But the problem remains: textbooks from the 1980s are still in use, and the design of the syllabus is a very serious problem. I recommend Yale’s Yihong Wu’s information theory lecture notes: www.stat.yale.edu/~yw562/ln.html

Introduction to Cybernetics

Teacher: Yang Xiaosong Credits: 4. Hours: 64 Textbook: Yang Xiaosong, Foundations of Cybernetics Evaluation: The content of this course is actually decent — it has depth, it is a serious test of your linear algebra, and it is fairly interesting. But I did not listen in class, because the first three weeks were spent telling big stories — feedback, feedback, feedback! I got a little tired of hearing it. Then I missed a week or two, and the teacher took off like a rocket, so I had to read on my own to catch up.

Stochastic Processes

Teacher: Wang Xiangjun Credits: 4. Hours: 56+8 Textbooks: Liu Cihua; Lin Yuanlie; Zdzislaw Brzezniak & Tomasz Zastawniak Evaluation: The breadth was fine — everything that ought to be covered was touched on, the only question being how deeply. The teacher’s technique was decent too: a quick review of the homework before class, and class time used efficiently. Reference books and supplementary material were sent to us promptly (which is very good), and the content of the computer lab sessions was relevant to the course. Overall the result was good. The book by Zdzislaw Brzezniak & Tomasz Zastawniak is also very good, and well suited to self-study!

Differential Geometry

Teacher: Zhang Ning Credits: 4. Hours: 64 Textbooks: Peng Jiagui; Chen Weiheng Evaluation: The breadth was fine, and the volume of computation is large, so it demands a fair amount of effort — but it really is a rare geometry course. I also fell a few weeks behind, could not keep up with the pace, and ended up reading on my own without attending.

Mathematical Modelling and Mathematical Experiments

Teacher: Xu Haoyuan Credits: 2 (Mathematical Modelling), 1.5 (Mathematical Experiments). Hours: 32+24 Evaluation: I do not think the lectures are necessary — there is not much theoretical depth, and you grasp it after a short look. It would be better if all of it were hands-on lab work. That is to say, drop the 2 credits of mathematical modelling as well; replacing it all with mathematical experiments would be fine too.

Xingyu Chen 25 April 2022