ma521: Representation Theory - w25

Daniel Rowe
darowe{at}nmu{dot}edu

I'm an associate professor of mathematics in the Mathematics and Computer Science Department at Northern Michgan University. I've been a professor at NMU for nine years, and I am very passionate about the praxis of doing mathematics and teaching it. I grew up on a fishing camp in Northwestern Ontario, Canada.

This graduate-level course offers an introduction to representation theory, a cornerstone of modern mathematics with deep connections to physics, geometry, and algebra. The course will cover the fundamental concept of a representations of group (finite or Lie-type), along with a review of linear algebraic notions such as subspaces, quotients, tensor products, Hom spaces, and symmetric and skew-symmetric tensors. Topics include the classification of simple and indecomposable representations, the representation theory of finite groups over the complex numbers, and the structural theory of simple Lie algebras over the complex numbers. Applications and connections to other areas of mathematics and physics will be highlighted throughout.

• Office Hours
• W 1-2, R 11-12, F 11-12
• JXJ 2228
• zoom link - passcode 809390
• read the relevant section(s) of our materials
• study all posted solutions
• re-watch the recorded lectures

• hybrid-flexible, in-person and over zoom
• strive for in-person attendance if possible!
• avoid becoming reliant on zoom and recordings!
• engagement is vital to learning mathematics (or anything)

• (40%) Homework
• (20%) Midterm Exam
• (15%) Research Paper (Posted to the Class)
• (25%) Final Exam

In the spirit of academic honesty, credit for this section is due to Asher Auel, as this is an adapted form of their discussion of academic honesty in mathematics.

• Working with others on mathematics, and using electronic resources is both highly encouraged and fun. You may work with anyone (e.g. classmates, non-classmates, tutors, etc.) If this is done well, you'll learn more effectively and efficiently.
  
  Here's the fundamental rule:

  Work with anyone or anything to develop your own personal understanding of the ideas required to solve your homework problem, but always write-up the final draft by yourself and in your own words.

• Writing up the final draft is just as important as figuring out the problems on scratch paper with your friends, using the internet, ChatGPT, etc. If you work with people, or use electronic resources on a particular homework:

You must list your collaborators and electronic sources at the top of the very first page. This makes the process completely transparent and honest.

A Note About Copying Mathematics

Mathematical writing is idiosyncratic; if your assignments are copied, it is quite easy to tell. You will not learn by copying solutions from others, or from external sources such as internet forums (e.g. math.stackexchange) and generative AI (e.g. ChatGPT). Regarding internet forums, you are free to look at them and use any understanding you've gained from them. Be warned that internet forums often contain incorrect or circuitous solutions, misleading discussions, use of techniques outside of the course material, and other material that may be detrimental to your learning process. Even the time that it takes to repeatedly search for solutions and read through dozens of forum posts could be better spent learning the material on your own or composing a question to the instructor or classmate. Regarding generative AI (e.g. ChatGPT), you are free to experiment with asking questions, but be warned that these systems are currently still very bad at deductive reasoning, and that the output may contain a mix of correct, incorrect, and unverified statements. Ask them to prove something false, they will work hard to do so, often giving contradictory answers. Therefore, I would be very careful with using these tools as learning resources on your own.

Punishments

In this modern world of online resources, cheating, copying, copy-and-pasting, ChatGPT, etc.; it is now more important than ever that citizens develop the intelligence to use their own brain to solve problems. I want my classes to be a postive force in this regard, by promoting the principles of academic honesty, and punishing those who disrespect those principles. The first infraction will result in a 0% on the entire assigment and a stern warning. If there is a second infraction, I will pursue sanctions through the Dean of Students office.

If you have a need for disability-related accommodations or services, please inform the Coordinators of Disability Services in the Dean of Students Office at 2001 C. B. Hedgcock Building (227-1737 or disability@nmu.edu). Reasonable and effective accommodations and services will be provided to students if requests are made in a timely manner, with appropriate documentation, in accordance with federal, state, and University guidelines.

• The main reference for the class will be my hand-written lecture notes (to be posted soon).


• These notes will combine bits and pieces from the following sources:


• William Fulton and Joe Harris, Representation Theory - A First Course, Springer, 2004.
• Pavel Etingof, Oleg Golberg, et. al., Introduction to Representation Theory, American Mathematical Society, 2011.
• David Bishop, Group Theory and Chemistry, Dover, 1973.
• Jean-Pierre Serre, Linear Representations of Finite Groups, Springer, 1977.
• S. Sternberg, Group Theory and Physics, Cambridge University Press, 1994.


• as well as Richard Borcherds' YouTube lecture series

• master list of homework problems


• hw1: Q1-Q5 (due 1/22 @ 11:59PM) → sol
• hw2 (due ## @ 11:59PM) → sol
• hw3 (due ## @ 11:59PM) → sol
• hw4 (due ## @ 11:59PM) → sol
• hw5 (due ## @ 11:59PM) → sol
• hw6 (due ## @ 11:59PM) → sol
• hw7 (due ## @ 11:59PM) → sol


• practice_midterm → sol
• midterm_exam → sol


• final_exam

• please put a .pdf file inside our shared google folder
• the shared google folder will be titled w25_ma521_yourlastname
• I will share it with you within first two weeks of class
• please don't submit anything via email attachment
• name your files in an organized manner, for example: hw1_Jane_Smith.pdf
• always show your work and keep it organized
• indicate/circle/highlight your answers
• answer the questions in the correct order