Last Thursday, I gave my first lecture in a graduate class of mathematics. There were three other students in that class, and the professor. All of the students were graduate students in Algebra. I was the sole person in Analysis. At first, this intimate setting was pretty daunting. I hadn’t taken the class last semester and the prof obviously didn’t like me being part of it¹.
Each week, a student gives the lecture. The lecture lasts 3 hours². Every three or four weeks, the students will give solutions to the exercises that we have to solve. It can be pretty intimidating at first. I was extremely nervous. It wasn’t the fact that I was doing a presentation. As a teacher, I am in front of a class day in day out. It was because we had to present the demonstrations and proofs, explain them while the prof asked us about them. If you didn’t know the proofs well, then you were screwed.
I was actually dreading giving my presentation, and on Thursday I hadn’t eaten all day. Commutative Ring Theory is a challenging subject. I have an interest in Algebra, but my main focus of study is Analysis, including Measure Theory and Differential Geometry.
This semester, I have also got a Complex Analysis class. In that class, we are studying Harmonic mappings in the plane³. There are two other students in that graduate class and we have just started using this book. It’s an interesting topic, which I am somewhat familiar with.
While I was uncomfortable and nervous because of the prof and the fact that he was quizzing me while I was doing the proofs, I think I managed to get through it. After two hours, we did some exercises and he spent some time explaining some subtleties in the proofs.
Graduate classes are nothing like undergraduate classes. In undergrad classes, you can prepare the subject matter without delving on every single detail. In graduate classes, you need to know what you are talking about. It’s a lot of work, but we do learn a lot more and faster than undergrads. Of that I have no doubt.
In the last few semesters, I had taken some classes that weren’t specifically for graduate students, so the format was one that I was used to. This semester, it’s completely different. Infinitely more challenging, but ultimately more rewarding.
I hope to pick my advisor this semester as well as my thesis subject. I want to start working on it this summer so I can get going on it. In mathematics, you need to present original research in your thesis. You can’t do case studies. In fact, there’s no winging stuff around in math. You either understand it or you don’t.
From my experience, in the Canadian universities, most graduate classes still had magisterial lectures. When one of the graduate students from France wanted us to do exercises in front of the class in an undergrad class, a lot of us were shocked because it mean that we had to have solutions to all of the problem sets. We had a lot of problem sets.
I remember that in some of the advanced undergrad classes, the students did some exercises in front of the prof, but not that much. It’s something that makes you think about how lessons are given in university.
The lectures that we give rely solely on the blackboard and chalk. We don’t use computers, we just use our notes, so a lot of time has to be spent fleshing out the proofs from the book we use, Sir Michael Francis Atiyah‘s Commutative Algebras⁴.
This book is deceptively thin and sparse, but it covers a lot of ground. A lot of the theory is discovered more when you work on the problem sets. You can find the solutions to the exercises here [PDF]⁵. Some of this is also covered in Algebra textbooks, like Hungerford’s Algebra.
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[¹]: He had asked me to drop the class because he didn’t feel comfortable with me in it. The class was composed of his own graduate students in Algebra.
[²]: It’s basically like teaching the class. The other students do it in Mandarin, but I do it in English.
[³]: In fact, we are using Peter Duren’s book, Harmonic Mappings in the Plane.
[⁴]: We use Introduction to Commutative Algebra by M.F. Atiyah.
[⁵]: The solutions are by Athanasios Papaioannou.
[†]: I will be giving lectures in Complex Analysis and Commutative Algebra this semester. Both classes will have 3-hour lectures.