Anatoly Fomenko

Algebra and Geometry Through Hamiltonian Systems

Hamiltonian systems are considered to be the prime tool of classical and quantum mechanics. The proper investigation of such systems usually requires deep results from algebra and geometry. Here we present several results which in some sense go the opposite way: the knowledge about the integrable system enables us to obtain results on geometric and algebraic structures which naturally appear in such problems. All the results were obtained by employees of the Chair of Differential Geometry and Applications in Moscow State University in 2011–2012.

Billiard Systems as the Models for the Rigid Body Dynamics

Description of the rigid body dynamics is a complex problem, which goes back to Euler and Lagrange. These systems are described in the six-dimensional phase space and have two integrals the energy integral and the momentum integral. Of particular interest are the cases of rigid body dynamics, where there exists the additional integral, and where the Liouville integrability can be established. Because many of such a systems are difficult to describe, the next step in their analysis is the calculation of invariants for integrable systems, namely, the so called Fomenko–Zieschang molecules, which allow us to describe such a systems in the simple terms, and also allow us to set the Liouville equivalence between different integrable systems. Billiard systems describe the motion of the material point on a plane domain, bounded by a closed curve. The phase space is the four-dimensional manifold. Billiard systems can be integrable for a suitable choice of the boundary, for example, when the boundary consists of the arcs of the confocal ellipses, hyperbolas and parabolas. Since such a billiard systems are Liouville integrable, they are classified by the Fomenko–Zieschang invariants. In this article, we simulate many cases of motion of a rigid body in 3-space by more simple billiard systems. Namely, we set the Liouville equivalence between different systems by comparing the Fomenko–Zieschang invariants for the rigid body dynamics and for the billiard systems. For example, the Euler case can be simulated by the billiards for all values of energy integral. For many values of energy, such billard simulation is done for the systems of the Lagrange top and Kovalevskaya top, then for the Zhukovskii gyrostat, for the systems by Goryachev–Chaplygin–Sretenskii, Clebsch, Sokolov, as well as expanding the classical Kovalevskaya top Kovalevskaya–Yahia case.

Chapter 4: Cohomology Operations

Let n, q be two integers and let π, G be two Abelian groups. We say that a cohomology operation\(\varphi\)of type (n, q, π, G) is given if for every CW complex X a map \(\varphi _{X}: H^{n}(X;\pi ) \rightarrow H^{q}(X;G)\) is given and is natural with respect to X, in the sense that the diagram

Chapter 1: Homotopy

Let X and Y be topological spaces. Continuous maps f, g: X → Y are called homotopic (f ∼ g) if there exists a family of maps h t : X → Y, t ∈ I such that (1) \(h_{0} = f,h_{1} = g\); (2) the map H: X × I → Y, H(x, t) = h t (x), is continuous.

Chapter 2: Homology

Lecture 12 Main Definitions and Constructions Besides the homotopy groups πn(X), there are other series of groups coresponding in a homotopy invariant way to a topological space X; the most notable are homology and cohomology groups, H n (X) and Hn (X).

Chapter 6: K-Theory and Other Extraordinary Cohomology Theories

K-theory emerged as an independent part of topology in the late 1950s, when the limits of the possibilities of the methods based on spectral sequences and cohomology operations (and studied in Chaps. III–V) became visible. Progress in homotopy topology considerably slowed down, the leading topologists got involved in cumbersome calculations, the results became less and less impressive, and to obtain them one had to combine a great inventiveness with a readiness to do a huge amount of tedious work. To get these activities revived a radical method was needed. And it was found! It consisted of replacing cohomology as the basic homotopy invariant by an entirely new object, the so-called K-functor.

Chapter 5: The Adams Spectral Sequence

As we demonstrated in the previous lecture, the information about the action of stable cohomology operations may be used for computing stable homotopy groups. If we know the cohomology of the space X, we can find, relatively easily, the “stable part” of the cohomology of the first killing space of X, and then the same for the second, the third, and so on killing spaces. Hurewicz’s theorem gives us, every time, the corresponding homotopy group. This method (the Serre method) does not permit us, however, to find the homotopy groups without dealing with other difficulties.