Richard Borcherd’s tips on doing mathematics

Portrait Richard Borcherds

Recently, I discovered Richard Borcherd’s Youtube channel. At the time when I discovered it (in February of 2021, I think) he was churning out new lectures almost daily, currently he’s posting a new lecture once per week. The topics of his videos are mathematical lectures on mostly graduate topics, but also undergraduate. I liked particularly his number theory playlist and his complex analysis playlist. The modular forms playlist was it bit hard for me to follow, I guess I wasn’t ready yet.

For following along the complex analysis playlist, I used mostly the book Funktionentheorie 1 by Freitag and Busam, which is in German. For the number theory lectures, I used Einführung in die algebraische Zahlentheorie by Schmidt, also in German.

For those of you who don’t know: Richard Borcherds has received many prizes throughout his career, one of which was the Fields Medal. I like his lecture style because it contains many examples and does not feel like someone is reading a text book to you. If I want the text book level of detail I can always go there and try to fill in my gaps. But with Borcherd’s lectures, one can easily get an overview of a topic, learn which concepts are really important and which examples are central to a thorough understanding.

On the 3rd of March, Richard Borcherds made a video which was a mathematical Q&A. I still remember very vividly, that I was quite excited about its anouncement a few days earlier. He had asked all his viewers about questions which they wanted to have answered and I posted some questions myself. For me, the Q&A video contained a few very helpful comments. First of all, he recommended all books and papers by Serre, in particular his book “A course in arithmetic”, and secondly, he explained that his style of learning mathematics consists mostly of “finding out what’s really going on” by calculating examples instead of memorizing proofs. This Q&A video really motivated me and I was really thankful for having it, even watched it twice. He really gave the impression that even an experienced fields-medal-winning established mathematicians has his struggles in one form or another. And we can be quite thankful for the fact that he sometimes struggles with mathematics, too, because I got the impression that this struggle gave us this wonderful Youtube channel.

By the way, if you want to learn more about Richard Borcherds, here’s an interview with him on Youtube and an article from 1998 by Simon Singh for the Guardian.


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