(CC BY-SA 4.0) : link
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.
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. Finite 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 concludes 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.
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.
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.
You must list your collaborators and electronic sources at the top of the very first page. This makes the process completely transparent and honest.
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.
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.