# Great graduate courses that went online recently

In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie Groups, and Modular forms! (and an undergraduate courses in Theory of numbers and Complex analysis).

I watched so far about 300 of his videos (about 90%) and they are really great. Borcherds is an amazingly good lecturer (for my taste). It is also clear, that these lectures were worked out/improved through years, since Borcherds was giving similar ones in Berkeley (one can find some lecture notes by students online). I would guess, that currently there are some other great lecturers that started to upload their courses on YouTube (or some other platforms). For this reason a question.

Question. If you watched recently an online graduate course (free for all), and found it brilliant, by a lecturer whom you find great, and believe that the course taught you something, could you please share the info about it.

The motivation for this question is to spread information about exciting things happening in mathematical life/education lately.

• You are right. As for 163, I guess, it is explained in this video: youtube.com/watch?v=a9k_QmZbwX8&t=4s As for 3264 - this is the number of conics in $\mathbb P^2_{\mathbb C}$ tangent to 5 given ones. And 248 - I don't know :) Jan 12 at 0:34
• 248 is the dimension of the largest exceptional Lie algebra $E_8$. Jan 12 at 1:49
• I stumbled on Borcherds' page around Christmas and was blown away by the quality of his videos (at the time I didn't know who he was) and have been wondering about how to subtly advertize for his stuff. Jan 12 at 11:19
• Richard Borcherds is "only" a Fields Medalist (1998). Jan 12 at 21:00

Algebraic Geometry in the Time of COVID (AGITTOC) happened during late summer of 2020. Ravi Vakil gave what he called "pseudolectures" following his "Rising Sea Notes". He didn't cover the whole book.

The blogposts, for some guidance through the exercises, and other thoughts:

He was planning on starting it up again in some capacity, at some point, but I don't know what happened to that plan. This gave some structure and purpose to what was otherwise a miserable and depressing summer for me, so I have a soft spot for it.

• There was also a Zulip chat for AGITTOC, which I believe people can still get access to one way or another (this should be outlined somewhere in the blog). It contains the discussion of the participants who were involved in AGITTOC, and is definitely worth looking at. Jan 12 at 5:47

Most of them are not so recent as you are asking for, but in my opinion is still worthy to look at them.

1. Claudio Arezzo's lectures on differential geometry https://www.youtube.com/playlist?list=PLp0hSY2uBeP_2dottSdX0eLPV_but7UAG

2. All the videos of Joe Harris are wonderful in my opinion, see: https://www.youtube.com/playlist?list=PLTWOVL6OX7hCIFAbbE4q0aSCrRv4aUxax, the Eilenberg lectures: https://www.youtube.com/playlist?list=PLj6jTBBj-5B_QE35IEQgLkkEct0Dk8GG6, the lectures on Poncelet's theorem https://www.youtube.com/watch?v=P7E_YOiuDvI

3. Robin Hartshorne's lectures on deformation theory: https://www.youtube.com/playlist?list=PLTWOVL6OX7hC2KrBaxkx_dzEpHyOJqMZm

4. All the lectures on Algebraic Geometry by Miles Reid, even if the audio/video quality is not the best. See for example: https://www.youtube.com/playlist?list=PLY28_CByrwHZ0UOqfuw6KVgKIE6-Go0RG

5. Pierre Albin's series on algebraic topology: https://www.youtube.com/playlist?list=PLpRLWqLFLVTCL15U6N3o35g4uhMSBVA2b

6. The channel of Federico Ardila for the lovers of combinatorics: https://www.youtube.com/user/federicoelmatematico/playlists

7. These lectures by Bernd Sturmfels on tropical geometry: https://www.youtube.com/playlist?list=PLRy_Pn1LtSpfYE39TuCZrJAi7-0z44sQB

9. Do not forget Norman J Wildberger's channel: https://www.youtube.com/user/njwildberger, it is very good, even if he thinks that infinity does not exist :)

10. Finally since I've fallen in love with Don Zagier, I suggest all of his lectures on youtube. If I had to advise a particular one, certainly the series at ICTP "Rogers-Ramanujan identities and the icosahedron" https://www.youtube.com/playlist?list=PLLq_gUfXAnkn64UkHxrM9wWZnBXy6xUM7

Let me finish citing some maybe less technical channels but still very enjoyable.

1. The Math doctor Bob series on Representation Theory: https://www.youtube.com/playlist?list=PL57457844458A5A1F
2. Billy Woods's channel on algebraic number theory: https://www.youtube.com/channel/UCcNQgs6bKds_KBpjp2_oavw/videos
3. If you love competition maths problems with also some insight in analysis Michael Penn's channel is a must https://www.youtube.com/channel/UC6jM0RFkr4eSkzT5Gx0HOAw
• Thanks a lot for this long list. I wonder, have you seen anything else good on Representation Theory (apart from doctor Bob)? Jan 15 at 23:39
• @aglearner yes there are lectures by Frederic Schuller (but maybe with a more physics flavour) and also the ones by Borcherds. I recommend also the brief course by Villegas available at the ICTP's youtube channel.
– gigi
Jan 18 at 14:26
• Michael Penn also has a great introductory series on vertex operator algebras, including a guest lecture from Borcherds. Jan 23 at 11:41

Yufei Zhao has a course titled "Graph Theory and Additive Combinatorics" available on MIT OpenCourseWare.

• This course is very well presented! At least the first lecture is well worth watching if you are even a little interested in this very charming field. Jan 12 at 7:46

Timothy Gowers on combinatorics.

Sean Prendiville on additive combinatorics. His lecture is somewhat scattered on the webinar page. He also provides lecture notes.

Frederic Schuller, theoretical physics, brilliant lecturer:

Not pure mathematics, I know, but brilliant introductions to some very advanced applications.

• I am currently finishing the geometric anatomy series and they are by far the greatest lectures in theoretical physics I have ever watched, teaching some extremely fundamental tools (topology, manifolds, fiber bundles, Lie groups/algebras, principal/associated bundles, and advanced applications in physics). I look forward to watching the other two series as well.
– Kai
Jan 12 at 23:46

I think Dev. Sinha's channel also is a great source for people interested in algebraic topology.

• People can see the whiteboards of my first lecture series at this page: pages.uoregon.edu/dps/GeometricAlgebraicTopology I hope to gather some more materials on a revamped version of my web page (e.g. expository papers, notes) before too long. Jan 14 at 19:46

Carl Bender's lectures on very powerful but non-rigorous perturbation theory methods used in theoretical physics.

• @Wojowu Thanks for the link to the playlist! Jan 12 at 20:57

IMPA has several courses online, some of them in English, including:

You can find all courses (including those in Portuguese) here.

Discrete Differential Geometry Keenan Crane, Carnegie Mellon University, Spring 2021.

]

Over the last term Jarod Alper has taught a course Introduction to stacks and moduli, introducing algebraic spaces and stacks and with the goal of proving (or outlining) properties of the stack $$\mathcal M_g$$ of curves, its compactification and their coarse moduli spaces. The recordings of the lectures are available and a pretty complete set of notes has been compiled.

The course only assumes prior familiarity with standard scheme theory, however a lot of other background material is covered in the appendices. One warning is that Jarod uses a bit of nonstandard terminology regarding stacks in general.

Two more recent ones (I think they are both still going on):

• Thanks for the references. Do you think that the first course is recorded? I can't find a link to the videos of lectures on the webpage of the course, only a zoom link. Jan 13 at 8:51
• Email them for access to the recordings. There is also a piazza. Jan 13 at 22:02

The lectures for Steven Strogatz's (ongoing) course on "Asymptotics and Perturbation Methods" are posted to his YouTube channel: https://www.youtube.com/user/stevenstrogatz1/videos. Strogatz is an outstanding lecturer.

Here’s a course on C*-algebras and compact quantum groups by in my opinion a genius lecturer https://youtube.com/playlist?list=PLq3E5oubNNoAZi6W7968tNkSnIGTt4hEn

He also has a course on functional analysis (see the HSE youtube channel where the above lectures are uploaded). It is different from the most because he takes more algebraic approach.

This course on Real Algebraic Geometry from Konstanz Universität's professor Markus Schweighofer is a really good introduction to the topic.

• Thank you all for these nice links. I was wondering if any who knows video lectures for advance real analysis or set theory
– 00GB
Jun 25 at 2:02

Ben Webster gave an introductory symplectic geometry course this past term which I think was very good. The presentation was pretty elementary. The course was offered through the Fields Institute, and their YouTube channel also has links to some other courses by some professors around Ontario which were offered in the same way.

Course on Metric Space by Professor Ben Green https://youtube.com/playlist?list=PL4d5ZtfQonW10k_fkbX-9Jj7HGAqRZbM3

• user264745, Thank you all for this nice link. I was wondering if you know video lectures for set theory
– 00GB
Jul 1 at 3:58