Mathematics 790, Fall 2025

Minicourse: Morse theory and Floer homology

Time: Mondays and Wednesdays, 1:25-2:40 pm (please note change in days! this minicourse no longer meets on Tuesdays and Thursdays)
Dates: September 29 to October 29
Location: Gross 324
Instructor: Lenny Ng

Morse theory has historically served as an important intermediary between differential geometry and algebraic topology, and it has gained further importance in topology in the last few decades through the work of Floer, Witten, and many others. This minicourse will provide a rapid introduction to Morse theory with the goal of defining Morse homology. We'll then pivot to Floer homology, which can be thought of as an infinite-dimensional analogue of Morse homology, and discuss what's different in this setting.

Topics I plan to discuss:

• Morse theory: Morse functions, surgery and handles, Morse inequalities, Morse-Smale pairs and gradient-like flows
• Morse homology: the Morse-Smale-Witten complex, orientations and signs, Poincare duality
• Floer homology: an overview of selected flavors, probably instanton Floer homology and symplectic Floer homology. (This will probably be very rapid because of time constraints.)

Everyone is welcome and I'll try to keep prerequisites to a minimum. It will be helpful, but not completely necessary, to have a basic familiarity with algebraic topology along the lines of Math 611, and smooth manifolds along the lines of Math 620.

Information sheet for the minicourse (basically just the information above)

Course notes are available for a previous version of this minicourse from spring 2015. See here for the web page for that minicourse, including lecture notes.