One of the main things that graduate students need to cope with in the sciences is giving lectures. For some, they give lectures to undergrads. Others give lectures in their classes. In my case, since I am not fluent in Mandarin, I can’t serve as TA, which is what most of my classmates have to do. However, in both of my classes, we have to give lectures. This is very different from giving a 30 min to 1h presentation. Students have to assimilate new subject matter and present it to the class, while the prof is watching. In both of my classes⁴, there aren’t that many students³.
This can be quite challenging because you need to prepare fully before you give a 3h lecture. While you give the lecture, the prof will ask you questions about the new topics and proofs, to see if you have understood it. He/She will ask to see if the other students have understood as well. In my case, in my Commutative Algebra class, most, if not all, of the explanations are in Mandarin, but I usually get what’s being explained since I tend to prepare the topics even when I don’t have to give the lecture.
This year, the graduate class format changed dramatically for me. I went from a normal class, filled with students, to classes with at the most 4 students and a professor. Actually, my Complex Analysis II class has only another student enrolled. As such, the format has changed. The professors no longer give 3h-lectures, the students do, each in turn.
Basically, each graduate student will prepare a 3h-lecture¹. In one class, that means that I lecture every 4 weeks. In another, it’s every other week². Preparing the lecture involves going over the textbook and the proofs. Depending on how detailed the proofs are, you’ll need to flesh them out further, and make them understandable, citing the right theorems, propositions, etc. Depending on what book/resources you are using, this might take quite a few hours. It also depends on the overall complexity of the class and the overall sparseness of the authors of the book. Atiyah’s books is very sparse. The proofs are sometimes quite short and they need to be expanded significantly.