ma521: Symmetry, Geometry, and Representations - f26

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 mathematics professor for eleven 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.

• Office Hours
• MW 12:45-2, R 11-12
• JXJ 2228
• zoom link - passcode 809390
• read our course materials
• study all posted solutions
• re-watch recorded lectures

• hybrid-flexible, in-person, and over zoom
• strive for in-person attendance
• avoid becoming reliant on zoom and recordings
• use them for extenuating circumstances only
• engagement is vital to learning mathematics (or anything)

• (35%) Homework
• (20%) Tests
• (15%) Final Project
• (30%) Final Exam

This graduate course develops a broad, geometric perspective on modern mathematics through the unifying theme of symmetry. Rather than beginning with a particular branch of mathematics, we ask how algebraic structures, geometric spaces, and representations arise naturally from one another. The course has five interconnected pillars. We begin with vector spaces and bilinear forms, developing the language of Euclidean, symplectic, orthogonal, and Hermitian geometry. We then explore the zoo of groups: discrete groups, Lie-type groups, Coxeter groups, and groups arising from geometry and number theory. Next, we study geometric spaces with symmetry, emphasizing homogeneous spaces and double quotients. Euclidean, spherical, and hyperbolic geometries emerge naturally from this perspective. We then turn to representations and harmonic analysis: groups act on spaces of functions, and we seek to decompose these actions into simpler modes. Fourier analysis, character theory, spherical harmonics, and cohomology provide increasingly sophisticated ways of extracting finite-dimensional representations from geometric spaces. Finally, we introduce Lie algebras as the infinitesimal theory of symmetry: derivatives of group actions, invariant vector fields, connections, and their curvature. Time permitting, this course will conclude with selected glimpses of geometric representation theory and modern mathematics, such as Borel Weil Bott, Springer theory, theta functions, and arithmetic representation theory. The goal is not mastery of a single technical theory, but the development of a unified mathematical language for symmetry, geometry, and representation.

• the instructor's job is to ensure the course content is relevant, impactful, clear, organized, and interesting


• your job is to attend as many classes as you can, engage your mind, ask questions, read, and budget (at least) 2-3 focused hours every week to work on the course content

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 2101 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.

• Here are my hand-written lecture notes for the semester.

None of the following books are required to complete the course. If you are interested in any of the subtopics we encounter, feel free to use these texts as supplementary material.


• Artin - Geometric Algebra
Excellent for the bilinear forms → symmetry transition.


• Hall - Lie Groups, Lie Algebras, and Representations
Introduction to groups of Lie-type and their Lie algebras.


• Humphreys - Reflection Groups and Coxeter Groups
A natural reference for Weyl/Coxeter groups.


• Fulton, Harris - Representation Theory: A First Course
Best overall supplement for the representation theory portions.


• Arvanitoyeorgos - An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
A natural reference for the symmetric space G/H philosophy.


• Cavalieri, Miles - Riemann Surfaces and Algebraic Curves
Exploring the relationship (galois group) ↔ (fundamental group).

• homework
• hw1 (Q1-Q2) due 9/4 @ 11:59PM - sol (posted 9/8 @ 3:00PM)
• hw2 (Q3-Q5) due 9/16 @ 11:59PM - sol (posted 9/21 @ 10AM)
• hw3 (Q6-Q8) due 9/27 @ 11:59PM - sol (posted 10/5 @ 9:00AM)
• hw4 (Q9-Q11) due 10/21 @ 11:59PM
• hw5
• hw6


• practice_test1 - sol
• test1 - sol
• test2


• final_project ideas
• 1) bilinear forms in characteristic 2
• 2) classification of bilinear forms over Q
• 3) conjugacy classes in various groups
• 4) translations, rotations, reflections, and glides in E(2)
• 5) Limits of RREFs in Grassmannian spaces Gr(k,n)
• 6) Flag Varieties Fl(d₁,...,d_k)= matrices up to B-operations
• 7) Schubert cells in G(k,n) and/or Fl(d₁,...,d_k)


• final_exam

• all tests and final exam will be traditional hand-written exams


• everything else must be submitted on-time as a SINGLE .PDF file to a shared google folder titled f26_ma521_lastname_firstname in an organized manner, for example hw1_Jane_Smith.pdf, etc.


• don't submit homework via email attachment
• always show your work and keep it organized
• circle or highlight your answers, if appropriate
• keep the questions in the correct order