Lectures: TTh 1:30-3:05pm; Online, zoom link to be emailed to participants

Instructor: Viktor Ginzburg; email: ginzburg(at)ucsc.edu,

Office Hours: TBA or by appointment

Prerequisites: Basic theory of manifolds and (co)homology and some elements of differential geometry

Text: There will be no “official” textbook in this course. I will go through some suggested reading in class.

Tentative Syllabus: Morse theory is one of the most useful methods in modern geometry and topology with applications to such diverse fields and algebraic topology, symplectic geometry and topology, algebraic and differential geometry, and calculus of variations. In fact, it is much more than a tool: it is a way of thinking about geometrical problems.
The theory, broadly understood, relates critical points of a function and the topology of its domain. For instance, in its classical finite-dimensional version, Morse theory gives in lower bound on the number of critical points of function on a closed manifold, under a minor non-degeneracy requirement, in terms of the homology of the manifold. The theory is used in both ways: to get information about critical points of a function from the topology of the space and also to study the topology of the space by using a function on it.

We will start this course with a discussion of the finite-dimensional Morse theory from a modern perspective, define the Morse complex and prove the Morse inequalities. To illustrate its power and usefulness we consider several application of the theory belonging to different areas of geometry: the Lefschetz hyperplane section theorem (algebraic geometry), the Bott periodicity (algebraic topology), and calculations of homology using Hamiltonian circle actions (symplectic geometry and algebraic topology). We will also discuss the “degenerate” variant of the Morse theory, the Lusternik-Schnirelmann theory. Finally, if time permits, we turn to infinite-dimensional versions of Morse theory and, e.g., the problem of existence of closed geodesics and other applications to differential geometry. Ideally, I would also like to touch upon Floer theory.

The course should be of interest to any student planning to study modern geometry and topology or already working in these fields.

The course will conclude with optional presentations by participants on selected topics related to Morse theory but not covered in class.

Lecture notes (pdf files)

• Table of Contents
• Week 1 (Lectures 1 and 2+): Generalities, the Morse lemma and all that homology
• Week 2 (Lectures 3- and 4): CW complexes and cellular homology, Morse inequalities, handles
• Week 3 (Lectures 5 and 6): the Floer theory perspective, examples and applications, towards existence and density of Morse functions (the transversality approach)
• Week 4 (Lectures 7 and 8): Jet transversality; existence and density of Morse functions (via height functions); products
• Week 5 (Lectures 9 and 10): Lusternik-Schnirelmann theory, LS category and the cuplength, Minimax Principle
• Week 6 (Lectures 11 and 12): Critical value selectors, the Courant-Fischer theorem, the Lusternik-Schnirelmann theorem on closed geodesics, the Palais-Smale condition
• Week 7 (Lectures 13 and 14): Closed geodesics; Cartan’s theorem and the Lusternik-Fet theorem (the broaken geodesics approach)
• Week 8 (Lectures 15 and 16): Cartan’s theorem and the Lusternik-Fet theorem – the infnite-dimensional approach, other calculus of variations problems (Benci’s theorem), geodesics connecting two points and the homology of the based loop space
• Week 9, Lecture 17: Applications: Bott’s periodicity and the Lefschetz hyperplane section theorem
• Week 9, Lecture 18: by Erman Cineli: Morse-Novikov theory
• Week 10, Lecture 19: by John Pelais: Lagrangian Floer homology
• Week 10, Lecture 20: by Elijah Fender: Morse-Bott and equivariant Morse homology

Lecture notes — the entire course without Lectures 18-20 (pdf file, 200MB)

Recordings of lectures

• Lecture 1, 01/05
• Lecture 2, 01/07
• Lecture 3, 01/12
• Lecture 4, 01/14
• Lecture 5, 01/19
• Lecture 6, 01/21
• Lecture 7, 01/26
• Lecture 8, 01/28
• Lecture 9, 02/02
• Lecture 10, 02/04
• Lecture 11, 02/09
• Lecture 12, 02/11
• Lecture 13, 02/16
• Lecture 14, 02/18
• Lecture 15, 02/23
• Lecture 16, 02/15
• Lecture 17, 03/02