In the early 1920s M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology.
Circle-valued Morse theory originated from a problem in hydrodynamics studied by S. P. Novikov in the 1980s. Nowadays, it is a constantly growing field of contemporary mathematics with applications and connections to many domains of geometry and topology.
The aim of this course is to give a systematic treatment of geometric foundations of the subject and recent research results.
Students will be able to understand basics of Morse theory and circle-valued Morse theory.
Morse theory, circle-valued Morse theory
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course
|Course schedule||Required learning|
|Class 1||This course covers the fundamentals of Morse theory and circle-valued Morse theory and several topics such as: - Homology with local coefficients for the Morse-Novokov theory; - The Morse-Novikov theory for closed 1-forms; - Circle-valued Morse theory for knots and links, etc.||Details will be provided in class.|
A. V. Pajitnov, Circle-valued Morse Theory, Walter de Gruyter.
Students require the knowledge of manifolds and homology.