Math 4010 & 5530 (Spring 2023) Homepage

News & Announcements

5/10
Final grades have been posted. I sent a final email grade report.

Also, just because this class is over, don't feel like you can no longer ask me questions. If you ever have a question mathematical or otherwise, please stop by my office or send me an email. I'm happy to help if I can! And finally, I hope you all have a fantastic summer break!

5/01
Here is my Roots & Weights Demo

Root Systems in Maple (.mw) This worksheet will plot Dynkin diagrams, weights for irreducible representations for simple Lie algebras of ranks 1, 2, and 3. It will also find Cartan matrices, simple roots, fundamental weights, highest long roots, and do Weyl group computations.

4/27
Final Project & Presentation Guidelines and Suggested Topics:

Instead of a Final Exam, we will have a final "project" of sorts. I would like everyone to select a topic somehow related to this class (I have suggestions listed below). This needs to be something that we didn't cover or at least didn't cover in detail.

  • Pick and topic and clear it with me (in person or via email).
  • Study your topic.
  • Create a handout. This should be at least one page front and back.
    • Your classmates are your audience. Build off of background from class.
    • Let us know where to go for further reading somewhere in your handout.
      [Cite any valuable resources - like websites and textbooks - that you found.]
    • You might want to give a brief historical blurb.
    • Do worked out examples help us understand your topic? If so, include a few.
    • Is your topic about a "big theorem"? Include its proof or a sketch of its proof.
  • During the final exam period, everyone will give a brief presentation of their topic. 6 people x 10 to 15 minutes = about 1.5 hrs. So 10 to 15 minutes a piece should be good. A little longer or shorter should be ok too.
  • The presentation could use slides (like tex or powerpoint) or just use the chalkboard and handout.
  • Several people can work on related topics. If so, you'll need to coordinate presentations and handouts. However, everyone should be creating their own unique handout (that work should be individual not group work).

My Fall 2020 MAT 2110 page has some sample slides.
Or Slides (.pdf) [Source: (.tex)] from a recent talk I gave.

Handout? Why not model after one of my monstrosities. For example:
Differential Algebra and Liouville's Theorem (.pdf) [Source: (.tex)]
Some differential Galois Theory (.pdf) [Source: (.tex)]

Suggested topics/ideas:
  • Quantum/particle physics connections
  • Lie Groups and/or the connection between a Lie group and a Lie algebra.
  • Types A, B, C, or D in depth
  • Leibniz algebras
  • Tensor Products and Clebsh-Gordon
  • Coxeter Groups (A Generalization of Weyl Groups)
  • Constructing the exceptional algebra G2
  • Exploring exceptional algebras
  • The universal enveloping algebra and PBW theorem
  • The infinite dimensional Heisenberg algebra
  • Formal calculus (background for vertex algebras)
  • Kac-Moody Lie algebras (generalized Cartan matrices)
  • Theory of solvable Lie algebras (like low dim. classification)
  • Theory or nilpotent Lie algebras (like low dim. classification)
  • Representation theory of sl3=A2 (especially weight multiplicities)
  • Ado's Theorem
  • Anti-commutators: Jordan Algebras
  • Jacob -
  • Gavin - Lie groups
  • Tyler - Tensors
  • Jason - Ado's Theorem
  • Sam - Symmetry Methods for solve DEs
  • Nic - Coxeter groups
4/05
Homework #7 (.pdf) [Source: (.tex)] is due Monday, April 17th.

3/27
Homework #6 (.pdf) [Source: (.tex)] is due Wednesday, April 5th.

Prison & Sophus Lie plus Wakimoto waxes poetic: Story Time (.pdf)

3/20
Homework #5 (.pdf) [Source: (.tex)] is due Monday, March 27th.

2/20
We decided to have Test #1 a week from this Friday. We will have an in class part (Friday, March 3rd) and a take home part (due date TBD).

2/13
Homework #4 (.pdf) [Source: (.tex)] is due Monday, February 20th.

2/03
Homework #3 (.pdf) [Source: (.tex)] is due Monday, February 13th.

1/27
Homework #2 (.pdf) [Source: (.tex)] is due Friday, February 3rd.

1/20
Homework #1 (.pdf) [Source: (.tex)] is due Friday, January 27th.

Handout: Checking on a basis [(.tex)]

1/03
The 4010 syllabus, 5530 syllabus, tentative schedule, & asULearn
have been updated.