General Information

Introduction to the fundamentals of numerical analysis that uses programming skills to test the theoretical concepts of the class.

Prerequisites

The introductory math sequence class (MATH 2210-2220, MATH 2230-2240, or MATH 1920-2940) and familiarity with proofs (a 3000-level math class would be beneficial).

Topics Covered

  • Classes of matrices, such as orthogonal, unitary, or symplectic matrices
  • Lie algebras
  • Exponential mapping
  • Representation theory

Workload

Weekly problem sets (3-6 hrs), two in-class prelims, a final paper, and a final exam. There was also an extra long optional homework to get bonus points. [Spring 2024]

General Advice

  • There were three textbooks for this course: Naive Lie Theory by Stillwell, Matrix Groups for Undergraduates by Tapp, and Lie Groups, Lie Algebras, and Representations by Hall. Most of the homework problems were taken Stillwell. However, it is not quite general or rigorous enough, so lectures eventually stopped following it that closely. My suggestion is to skim Stillwell to do the homework (it’s pretty readable), but make sure to also keep up with lecture and to consult the other textbooks. At a certain point in the semester, Barbasch seemed to pivot to using Tapp, which is fine as a textbook, though beware it uses some different conventions/notations (ex. everything is written in terms of right actions). It worked for some people but I personally did not read it that much. The best textbook of the three is the one by Hall. It essentially covers the entirety of Stillwell in the first 70 pages or so, and it does so in a very cohesive manner. I highly recommend reading it; it is concise, clear, and general, and it eventually goes into some more advanced concepts that make for good topics for the independent study project at the end of the semester. [Spring 2024]

Testimonials

As a physics student, I found matrix groups to be a highly useful course. It helped formalize many mathematical concepts I had previously encountered in physics courses (ex. group theory in particle physics), and it was clear that many other physics students were interested in the topics as well–at least half of the class consisted of physics/AEP majors. However, it did not quite live up to my expectations, as we did not quite cover representation theory in class. That being said, it gave the background to learn it independently, and thus many students opted to explore representation theory for the final paper at the end of the semester.

In terms of specifics on course mechanics and such, Barbasch’s lectures could be hard to follow (ex. he would start a proof, then introduce a few definitions/theorems in between for 20 minutes, and then come back and finish the proof without indicating what he was doing), and the lecture notes posted on Canvas were not that clear to read either. Some weeks, it also felt like not much was covered. However, a combination of lectures and reading the textbooks should be enough to put the pieces together. Barbasch was also always willing to meet with students after class and during office hours. He’s a nice person and wants to help, just a little unclear sometimes.

Homeworks were a mix of computation and proofs, and the proofs were usually not too abstract or inaccessible. Occasionally, the homeworks would assume facts from analysis, algebra, or topology as given, and they could seem to come out of nowhere in the context of the course, so it was definitely helpful to have had background in some other math first before taking this class. Office hours and working with other students were other ways to fill in the gaps too.

The exams were generally pretty fair; there would always be a question or two that would stump everyone, but everything else was reasonable. This was a relief, given that the content could seem daunting while studying (ex. trying to remember all of the maximal tori of the different groups). Grading was generous as well. Barbasch also assigned an optional homework at the end that students could do for bonus points, which was very helpful for grades and for studying.

Overall, I feel that this course, while it did not quite live up to its potential (in my opinion), is still worth taking just because of how useful it is. Rating: 3/5. [Spring 2024]

Past Offerings

Semester Professor Median Grade Syllabus
Fall 2020 Birgit Speh B  
Spring 2024 Dan Barbasch A-/A? MATH4500_SP24.pdf