Science in English (Mathematics2)

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Andrei Pajitnov 
Lecture1  Exercise0  Experiment0
Syllabus updated
Lecture notes updated
Fall Semester

Outline of lecture

1) Novikov complex for circle-valued Morse functions.
2) Witten-type de Rham framework for Novikov inequalities.
3) Morse-Novikov theory for 3-knots
4) Applications to symplectic topology.

Classical Morse theory establishes a relation between the number
of critical points of a real-valued function on a manifold and
the topology of the manifold.

Circle-valued Morse theory is a branch of the Morse theory.
It originated from a problem in hydrodynamics studied by
S. Novikov in the early 1980s.
Nowadays it is a constantly growing field of geometry with
applications and connections to many geometric problems
such as Arnold's conjectures in symplectic topology,
fibrations of manifolds over the circle, dynamical zeta
functions, and the theory of 3-dimensional knots and links.

Purpose of lecture

Plan of lecture

Our course will start with recollections on the classical Morse theory
(gradient flows, Morse complex, Morse inequalities).
Then we proceed to the circle-valued Morse theory and construct
the Novikov complex (the generalization of the Morse complex to
the case of circle-valued functions).
This chain complex is generated over the Laurent series ring
by the critical points of the circle-valued function.

E. Witten showed in the beginning of 1980s that the classical Morse
theory can be reformulated in the framework of the de Rham theory.
We will explain his work, and its generalization to the Morse-Novikov
theory, which leads to a relation between the Novikov homology and
the homology with local coefficients.

A natural application of the circle-valued Morse theory is to the topology
of knots. If a knot K in the 3-dimensional sphere S is not fibred, then
any circle-valued function on S-K has critical points. The minimal number
of these critical points is an invariant of K, called the Morse-Novikov
number MN(K).
We will show how to give computable lower bounds for MN(K) with the help
of the Novikov homology, and relate it to the tunnel number of the knot.
In the last part of the course we will explain how the Novikov ring appears
in the Floer's work on the Arnold conjecture concerning the closed orbits
of periodic Hamiltonians, and discuss some recent developments in this direction.

Textbook and reference

Related and/or prerequisite courses


Mentioned in the class.

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