Faculty of Science Course Syllabus
Department of Mathematics & Statistics

MATH 4057/5057: Lie Theory (Winter 2026)

Instructor: Theo Johnson-Freyd, theojf@dal.ca.
Lectures: Mondays, Wednesdays, and Fridays, 15:35-16:25 Atlantic Time, Chase 227.
Office Hours: Thursdays, 14:05-16:55 Atlantic Time, Chase 214.
Course website: https://categorified.net/26Winter5057/.

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Course description

A Lie group is a group (a collection of symmetries) which is also a manifold (a smooth space). Lie groups arise whenever an object has continuous families of automorphisms. Finite groups are notoriously hard to classify. This is because groups exist in order to act on things, but a generic finite group may not have any easily accessible actions. Every Lie group, on the other hand, has a nice small linear space that it act on: its tangent space at the identity. This linear space furthermore comes with an important algebraic structure: the first derivative of the group multiplication. This algebraic structure makes it into a Lie algebra.

A nontrivial application of the theory of ODEs says even more: the Lie algebra of a Lie group almost completely determines the Lie group. This makes the analysis and classification of Lie groups much more accessible than the analysis and classification of discrete groups. In fact, for compact connected Lie groups, the analysis and classification reduces to a completely solved combinatorial problem. The goal of this course is to explain this classification.

Course materials and delivery

The main textbook for the class is:

• Compact Lie Groups by M.R. Sepanski.

Homework assignments and other materials will be available on the Course Content page.

This course will be offered in hybrid format. Lectures will primarily be delivered in person but broadcast over Zoom; the occasional lecture will be delivered over Zoom. https://pitp.zoom.us/j/94295219831

Course Assessment

There will be seven problem sets in the course, one every 1.5-2 weeks. The problem sets are designed to be hard and to take a long time: prepare accordingly. I encourage you to work together to complete your problem sets. Studies have shown that social ties are a main predictor of success in STEM classes. Although you are encouraged to work together, the homework you submit must be written by you individually.

There are many online resources offering solutions to homework at all levels. If you choose to use such resources, please be cautious: they often provide too detailed an answer, and students can trick themselves into thinking that they understand more than they do by copying those answers.

Rather than traditional exams, the final for the course will consist of a roughly 20-minute interview with the instructor during the exam period. Each student will select one problem from each problem set and submit their list ahead of time. During the interview, the instructor will select one problem from that list and the student to present the solution.

Additionally, each graduate students enrolled in Math 5057 will deliver a guest lecture sometime during the term, on a topic selected in consultation with the instructor.

Conversion of numerical grades to Final Letter Grades follows the Dalhousie Common Grade Scale.