Tag: group theory

Groups, Fields, and Galois: Part 1 | UCLA Extension

Dr. Michael Miller, a retired researcher at RAND, has been teaching upper level undergraduate/graduate level math courses for fun at UCLA Extension for over 50 years. This fall he’ll be introducing the areas of groups, fields, and Galois theory from abstract algebra to those interested in abstract math: Groups, Fields, and Galois: Part 1. His intention is to do a follow-on class on Galois theory in the Winter which will use this class as a foundation. If you’re in the Los Angeles area his class starts on September 22, 2026 at UCLA on Tuesday nights from 7-10PM. Register here: https://www.uclaextension.edu/sciences-math/math-statistics/course/groups-fields-and-galois-part-1-math-9001

His courses are thorough and rigorous, but geared toward lifelong learners and beginners in abstract mathematics to allow people better entry points into higher level mathematics. His classes are interesting and relatively informal, and most students who take one usually stay on for future courses. The vast majority of students in the class (from 16-90+ years old) take his classes for fun and regular exposure to mathematical thought, though there is an option to take it for a grade if you like. There are generally no prerequisites for his classes, and he makes an effort to meet the students at their current level of sophistication. For this particular class, if you’ve got some experience in high school algebra and know some preliminaries about mathematical proofs you’ll be ready to dive in.

There are regular commuters joining from as far out as Irvine, Ventura County and even Riverside. Most in the class are dedicated hobbyist and professional mathematicians, engineers, physicists, and others from all walks of life (I’ve seen actors, directors, doctors, artists, poets, retirees, and even house-husbands in his classes.)

If you’re unsure of what you’re getting into, I recommend visiting on the first class to consider joining us for the Fall quarter. Sadly, this is an in-person course. There isn’t an option to take this remotely or via streaming, and he doesn’t typically record his lectures. I hope to see all the Southern California math fans next month!

Course Description

Group theory is recognized as one of the great unifying mathematical disciplines, with far-reaching applications in virtually all scientific (and many non-scientific) fields. This course, the first of a two-quarter sequence that concludes in Winter 2027 with a course on fields and Galois Theory, is geared toward developing the tools of group theory needed to establish the Abel-Ruffini theorem on the insolvability by radicals of the general polynomial of degree five or greater. Results established along this journey will also be applied toward addressing the three classical ruler and compass construction problems of antiquity: trisecting the angle, duplication of the cube, and squaring the circle, as well as Gauss’ surprising characterization of constructible regular polygons.
September 22 – December 1
Tuesday 7:00PM – 10:00PM PT
REG# 410697
Fee: $450.00

Recommended textbook: TBD 

Dr. Miller provides enough background and thorough notes that if you’re taking notes on his lectures, you typically won’t need a textbook.

If you’ve never joined the class before (Dr. Miller has been teaching these for 53 years and some of us have been with him for nearly that long; I’m starting into my 20th year personally), I’ve written up some tips and hints.

Quantum mechanics anyone? Dozens have been disappointed by UCLA’s administration ineptly standing in the way of Dr. Mike Miller being able to offer his perennial Winter UCLA math class (Ring Theory this quarter), so a few friends and I are putting our informal math and physics group back together.

We’re mounting a study group on quantum mechanics based on Peter Woit‘s Introduction to Quantum Mechanics course from 2022. We’ll be using his textbook Quantum Theory, Groups and Representations:An Introduction (free, downloadable .pdf) and his lectures from YouTube.

Shortly, we’ll arrange a schedule and some zoom video calls to discuss the material. If you’d like to join us, send me your email or leave a comment so we can arrange meetings (likely via Zoom or similar video conferencing).

Our goal is to be informal, have some fun, but learn something along the way. The suggested mathematical background is some multi-variable calculus and linear algebra. Many of us already have some background in Lie groups, algebras, and representation theory and can hopefully provide some help for those who are interested in expanding their math and physics backgrounds.

Everyone is welcome! 

Yellow cover of Quantum Theory, Groups and Representations featuring some conic sections in the background

👓 Andrew Jordan reviews Peter Woit’s Quantum Theory, Groups and Representations and finds much to admire. | Inference

Read Woit’s Way by Andrew Jordan (Inference: International Review of Science)
Andrew Jordan reviews Peter Woit's Quantum Theory, Groups and Representations and finds much to admire.
For the tourists, I’ve noted before that Peter maintains a free copy of his new textbook on his website.

I also don’t think I’ve ever come across the journal Inference before, but it looks quite nice in terms of content and editorial.

🔖 Group Theory Lectures by Steven Roman

Bookmarked Playlist of Group Theory Lectures by Steven Roman by Steven Roman (youtube.com)
Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube.
Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube. No prior experience in group theory is necessary. He’s the author of the recent Fundamentals of Group Theory: An Advanced Approach. [1]

He hopes to eventually also offer lectures on ring theory, fields, vector spaces, and module theory in the near future.

Fundamentals of Group Theory by Steven Roman

References

[1]
S. Roman, Fundamentals of Group Theory: An Advanced Approach, 2012th ed. Birkhäuser, 2011.

🔖 Free download of Quantum Theory, Groups and Representations: An Introduction by Peter Woit

Bookmarked Final Draft of Quantum Theory, Groups and Representations: An Introduction by Peter Woit (Not Even Wrong | math.columbia.edu)
Peter Woit has just made the final draft (dated 10/25/16) of his new textbook Quantum Theory, Groups and Representations: An Introduction freely available for download from his website. It covers quantum theory with a heavy emphasis on groups and representation theory and “contains significant amounts of material not well-explained elsewhere.” He expects to finish up the diagrams and publish it next year some time, potentially through Springer.

I finally have finished a draft version of the book that I’ve been working on for the past four years or so. This version will remain freely available on my website here. The plan is to get professional illustrations done and have the book published by Springer, presumably appearing in print sometime next year. By now it’s too late for any significant changes, but comments, especially corrections and typos, are welcome.

At this point I’m very happy with how the book has turned out, since I think it provides a valuable point of view on the relation between quantum mechanics and mathematics, and contains significant amounts of material not well-explained elsewhere.

Peter Woit (—), theoretical physicist, mathematician, professor Department of Mathematics, Columbia University
in Final Draft Version | Not Even Wrong

 

Peter Webb’s A Course in Finite Group Representation Theory

Bookmarked A Course in Finite Group Representation Theory by Peter WebbPeter Webb (math.umn.edu)
Download a pre-publication version of the book which will be published by Cambridge University Press. The book arises from notes of courses taught at the second year graduate level at the University of Minnesota and is suitable to accompany study at that level.

“Why should we want to know about representations over rings that are not fields of characteristic zero? It is because they arise in many parts of mathematics. Group representations appear any time we have a group of symmetries where there is some linear structure present, over some commutative ring. That ring need not be a field of characteristic zero.

Here are some examples.

  • […]
  • In the theory of error-correcting codes many important codes have a non-trivial symmetry group and are vector spaces over a finite field, thereby providing a representation of the group over that field.”
Peter Webb, February 23, 2016, Professor of Mathematics, University of Minnesota
in A Course in Finite Group Representation Theory to be published soon by Cambridge University Press