**COVID-19 information:**You can as always email problem sets to me dav@math.mit.edu to arrive (at least a few minutes) before class time; I need time to print them for the grader to pick up immediately after class.Beginning Wednesday March 11, and continuing from Monday March 30, the class will be streamed. I plan to record the sessions and post them to this web page later in the afternoon.

**Records of class**: I am for now posting videos here. When I tried OpenBoard, I was able to export reasonable pdf's and put them here; but OpenBoard crashed constantly, so I wanted to try something else. I tried Microsoft OneNote. As far as i can tell this is unable to export reasonable pdf's; so for now I will put here a link which should allow you to view the notebook for each class.

If any of these methods creates difficulties for you, PLEASE tell me about them; perhaps I can suggest a solution, but certainly I can try a different method.Because of the cancellation of classes 3/13, I will

**NOT**record a class on Friday 3/13.

- notes (B) about the class and
**syllabus**from March 30. These notes will be updated from time to time and acquire a new letter (C, D,...)

**Meeting time:**Monday/Wednesday/Friday 3:00-4:00, NO LONGER in Room 2-142**Text:**Alexander Kirillov, Jr.,*An Introduction to Lie Groups and Lie Algebras.*- A beautiful older text explaining perfectly the dictionary between Lie groups and Lie algebras is Frank Warner's
*Foundations of differentiable manifolds and Lie groups*. An elegant treatment of the structure theory for Lie algebras is James Humphreys'*Introduction to Lie algebras and representation theory*.

David Vogan, 2-355

Telephone x3-4991

Zoom office hours Monday 4:00-5:00 **including 4.20** and Tuesday 2:30-3:30 (or any time; email to set up).

- video from 3/11/20
**NO MEETING**(online or physical) Friday 3/13/20.- video from 3/30/20. Math starts about 5:00.

Slides for class 3/30/20 (mildly updated 3/30 1pm). - video from 4/1/20. Math starts about 2:00.

Slides for class 4/1/20.

I promised an online proof of the Frobenius theorem. There are a ton. A nice one is by Chern and Wolfson, appearing as a three-page chapter in the Birkhauser book "Manifolds and Lie groups." (I picked that one because reading work by somebody like Chern is always a good idea.) I prefer that you should access it yourself through the MIT library rather than that I should (illegally!) post the pdf here. If you have difficulty finding it, please let me know. - OneNote notebook for classes 4/3/20 onward
- pdf file for notes kindly provided by Merrick Cai, because Microsoft OneNote won't read the OneNote file on a number of devices.
- video from 4/3/20. Math starts about 2:30.
- video from 4/6/20. Math starts about 5:00.
- video from 4/8/20.
- video from 4/10/20. Math starts 1:30.
- video from 4/13/20.
- video from 4/15/20.
- video from 4/17/20.
- video from 4/22/20.
- video from 4/24/20.
- video from 4/27/20.
- video from 4/29/20.
- video from 5/1/20.
- video from 5/4/20.
- video from 5/6/20.
- video from 5/8/20.
- video from 5/11/20.

Problem sets will be posted here.
Grading from March 15 will be on Gradescope, accessible at
GRADESCOPE. Registered students should already be on the class roster, with your Registrar name and MIT email address. If you have difficulty accessing this, let me know. PDF files of your solutions (which can be made from a photo of handwritten solutions if necessary) should be submitted on Gradescope by 16:00 Eastern Time on the due date (usually a Wednesday). The grader will mark the pdf files. Usually I hope to post solutions ** immediately after class Wednesday**.
This is the reason for the submission deadline. **No
extensions and no exceptions.** The grading for the course
will emphasize your best work, so missing one problem set should not
have a serious effect. (In light of the emergency P/No Record grading, you would need to miss quite a few problem sets to have difficulty.)

For Problem Set 5, the grader has everything that was submitted by March 11. I will put on Gradescope files emailed to me after that date; but if _you_ can submit them to Gradescope, that will be very helpful.

**First problem set due in class Wednesday,
February 12.**
**First problem set solutions, ** to be posted after class 2/12.

**Second problem set due in class Wednesday, February 19.**
**Second problem set solutions, ** to be posted after class 2/19.

**Third problem set due Wednesday, February 26.**
**Third problem set solutions, **to be posted after class 2/26.

**Fourth problem set due Wednesday,
March 4.**
**Fourth problem set solutions. **

**Fifth problem set due Friday,
March 20** (nine day extension).
**Fifth problem set
solutions, **to be posted Friday evening 3/20.

**Sixth problem set CANCELED**.

**Seventh problem set due Friday,
April 3.** (An problem set from an old course was inadvertently
posted here until 3/17/20. That is **not** the actual Problem set Seven, which is now posted here.

The due date printed on the pset is Friday, April 3; the web site until 3/28 said Wednesday, April 1, as had been scheduled since the beginning of the semester. I apologize that I did not put on the web site the extension that I intended and wrote on the pset. Future psets will be due on Wednesdays as usual.

**Seventh problem set
solutions, **to be posted after class 4/3.

**Eighth problem set due Wednesday,
April 8.**
**Eighth problem set
solutions, **to be posted after class 4/8.

**Ninth problem set due Wednesday,
April 15.**

The following slides concern Problem Set 9; they are just scratchwork from office hours, and not likely to be
useful if you weren't there (or perhaps even if you were).

Slide 1 from office hours 4/13.

Slide 2 from office hours 4/13.

Slide 3 from office hours 4/13.

Slide 4 from office hours 4/13.

Slide 5 from office hours 4/13.

Slide 6 from office hours 4/13.

Slide 7 from office hours 4/13.

Slide 8 from office hours 4/13.
**Ninth problem set
solutions, **to be posted after class 4/15.

**Tenth problem set due Wednesday,
April 22.**

The version of the problem set posted from Thursday night to Friday 17:00 contained an incorrect
link for the file roots.pdf referenced in the problems. The link is fixed in the
problem set now, or written here.
**Tenth problem set
solutions, **to be posted after class 4/22.

**Eleventh problem set due Wednesday,
April 29.**
**Eleventh problem set
solutions, **to be posted after class 4/29.

**Twelfth problem set due Wednesday,
May 6.**
**Twelfth problem set
solutions, **to be posted after class 5/6.

Not covered in 2020 are these notes on Weyl and Clifford algebras. I will nevertheless be very grateful for corrections or suggestions about improvements.

Here is ** a set of notes on root systems**, related to Chapter 7 of the text. This provides additional details on material I'll discuss in class at the end of the semester.

The notes that follow are older and not (so far) related to the 2015 course. Here are notes on compact classical groups.

Notes about actions of a Lie group on a manifold. You should also remember that the group SL(2,R) by its definition acts on the two-dimensional vector space R^2. Is there a way to get out of that an action on a circle?

Notes about quaternionic matrix groups. Goal is to relate quaternionic groups to complex groups. Edited Wednesday afternoon 11/5/14 to remove almost all coordinates.