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.