Lecture Series by Andrei Pajitnov

Morse theory is one of the most used tools of topology and geometry of manifolds. It establishes a relation between the number of the critical points of a smooth real-valued function on a manifold and the topology of the manifold. This relation was discovered by American mathematician Marston Morse around 1927, during his studies of variational problems. Morse himself was mainly interested in the applications of his theory to the case of infinite-dimensionall manifolds. However the time has shown that the Morse theory gives powerful insight into the structure of finite-dimensional manifolds as well.

Let us mention just two of innumerable applications of the theory. Smale's theory of handle decompositions, which is based on Morse-theoreticway of viewing themanifolds led to the proof of the Poincare conjecture in dimensions at least 5. An infinite-dimensional version of this theory includes the proof of existence of infinite number of geodesics joining two given points of a compact Riemannian manifold without boundary.

Circle-valued Morse theory is a new branch of the Morse theory. It originated from a problem in hydrodynamics studied by S. P. Novikov in the early 1980s. Nowadays it is a constantly growing field of contemporary mathematics with applications andconnections to many geometricproblems such as Arnold's conjecture in the theory of periodic orbits in Hamiltonian dynamics, fibrations of manifolds over the circle, dynamical zeta functions, and the theory of knots and links in three-dimensional sphere.

The course will consist of ten lectures and, time permitting, will cover the following topics.

Classical Morse theory: Introduction: manifolds that admit a function with only two critical points. Critical points and attaching of handles. Main theorem of the Morse theory. Geometric applications. Transverse gradients and the Morse complex. Witten's de Rham framework for the Morse theory.

Circle-valued Morse theory and Novikov homology: Introduction: circle-valued Morse functionsand Novikov inequalities. Construction of the Novikov complex. Rationality theorem and Novikov exponential growth conjecture.