Registration Deadline: January 19th, 2025

Instructor: Professor Andrew Lewis, Queen's University

Course Dates: January 6th - April 2nd, 2025
Mid-Semester Break: February 17th - 21st, 2025
Lecture Times: Mondays & Wednesdays | 9:00 AM - 10:30 AM (ET)
Office Hours: By arrangement with instructor

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

Capacity Limit: N/A

Format: Hybrid synchronous delivery

• In-Person (for students in Kingston)
• Online via Zoom (for students not in Kingston)

Course Description

This is an introductory course in infinite-dimensional geometry, i.e., the geometry of manifolds modelled on infinite-dimensional locally convex topological vector spaces, especially those that are not normed vector spaces. After an introduction to locally convex topological spaces, there follows a comprehensive introduction to differential calculus (the so-called Bastiani calculus) in these spaces; the standard topics include the Mean Value Theorem, the Chain Rule, class Ck-mappings and their properties. After an illustration that calculus in locally convex spaces looks a lot like ordinary calculus, two situations where this is not the case are discussed: the Inverse Function Theorem and ordinary differential equations. Using the material on calculus, the theory of manifolds modelled on locally convex spaces is covered; the standard topics here include manifolds, mappings, the tangent bundle, vector bundles. A main application of infinite-dimensional geometry is to spaces of mappings between manifolds, and this is covered in detail. Time permitting, the course will conclude with an introduction to infinite-dimensional Lie groups.

Prerequisites: While neither are strictly necessary, a student will probably benefit from some knowledge of finite-dimensional differential geometry and of infinite-dimensional normed vector spaces. But an advanced real analysis course is necessary.

Course Expectations: Anyone receiving credit will be expected to attend most lectures. Auditors are permitted.

Evaluation: There will be 4-5 assignments.

Textbook: I will provide detailed course notes drawn from a draft text by Helge Gloeckner and Karl-Hermann Neeb.