Instructor: Professor Hadi Salmasian, University of Ottawa

Course Dates: September 4th - December 2nd, 2024
Mid-Semester Break: October 14th - 18th, 2024
Lecture Times: Mondays, 4:00 PM - 5:20 PM | Wednesdays, 2:30 PM - 3:50 PM (ET)
Office Hours: TBA 

Registration Fee: PSU Students - Free | Other Students - CAD$500

Capacity Limit: 35 students

Format: Hybrid synchronous delivery

• In Person (for uOttawa students)
• Online via Zoom (for non-uOttawa students)

Course Description

The goal of the course is to first cover the foundational theory of Lie groups and then move on to more advanced topics that expose the audience to areas of active research. The following is the list of topics that are intended to be covered:

• Foundational theory of Lie groups: Lie groups, the exponential map, Lie correspondence. Homomorphisms and coverings. Closed subgroups. Classical groups: Cartan subgroups, fundamental groups. Manifolds. Homogeneous spaces. General Lie groups.
• Introduction to quantization: Symplectic manifolds, pre-quantization, the orbit method. Poisson manifolds, Manin triples. Universal enveloping algebras, quantum sl(2) and its representations, quantum symmetric spaces.

Course Texts:

The lectures on the foundational theory of Lie groups will be based on the following textbook:

• Hall, B., Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. 2nd Edition, Springer, 2015.

To download the book free of charge when you are connected to the internet on campus, click here.

The more advanced material on quantization will be covered from various references, including the following reference:

• Chari, V., Pressley A., A Guide to Quantum Groups, Cambridge, 1995.

Rationale: Despite its wide span of applications, the theory of Lie groups is not the subject of a standard graduate course offered by many math departments. At uOttawa, this course attracts graduate students working in both pure and applied mathematics, and we expect that it will serve the same purpose in the Fields Academy. The advanced part on quantization will be about 20% of the course content. This part will be adjusted according to the strength and interests of students that will register for the course.

Prerequisites: Strong background in 2nd/3rd year level algebra and real analysis is required. In particular, students should be familiar with basic group theory (e.g., normal subgroups, quotients, Lagrange’s theorem, isomorphism theorems, characterization of finite abelian groups) and elementary analysis (e.g., metric spaces, compactness, Heine-Borel Theorem, uniform convergence, series of functions).

Evaluation: There will be 4 assignments, 1 midterm exam, and 1 final exam. The exams will be in-person for uOttawa students, and will be taken on-line by students from other institutions. The midterm exam will be given during a lecture time interval, and the final exam will be 3 hours long.

For more information on course description, evaluation method (e.g., weight distribution), and course expectations, please refer to the course website: https://mysite.science.uottawa.ca/hsalmasi/MAT4144/mat4144.htm.