Winter 2026 Term

The purpose of the AARMS Advanced Course program is to give graduate students and upper year undergraduates from AARMS Member Universities an opportunity to take courses not offered at their home institution free of charge. In Winter 2026, the courses listed below will be offered in an exclusively online format.

Students may be able to claim credit for AARMS Advanced Courses from their home institution using mechanisms similar to the AARMS Summer School. Students hoping to receive academic credit for courses in this program are strongly encouraged to consult with their home institution about this process before the start of lectures.

Both students and instructors should consult the frequently asked questions.  Please send any addition questions about this program to Andrew Irwin.

Important dates

  • Applications will be accepted on a rolling basis

Registration

To register for an AARMS Advanced Course in Winter 2026 please contact the instructor directly.  Send a statement of interest to the instructor listed below indicating why you wish to take the class and listing previous courses relevant to the topic.

Lie Theory

Dr. Theo Johnson-Freyd, Dalhousie University

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.

The main textbook for the class is:

  • Compact Lie Groups by M.R. Sepanski.
Lectures: Mondays, Wednesdays, and Fridays, 15:35-16:25 Atlantic Time, Chase 227 or Zoom.