Beginning from Laplace and until comparatively recent time it was believed that the unpredictability in the classical mechanics is due to the errors in the initial data (even Max Born thought so, 1955). Of course, it is well known that the chaotic behavior of the classical systems has place at the random initial conditions, random external disturbances, excitations of the large numbers of the degrees of freedom. But these factors are only the sufficient and not the necessary reasons of the chaos.
Today we know a lot of simple low degrees of freedom (n ≥ 2) completely deterministic (I. e. described by the differential equations) systems with the extremely irregular, completely unpredictable motion (deterministic chaos).
I would like to speak about these extraordinary properties of the classical mechanics on the basis of the famous gauge fields.
Quantum chaos, quantum counterpart of the classical one, at the first glance should not exist due to the discreteness of the energy levels, essential linearity of the quantum mechanics, its uncertainty relation and the discreteness of the phase space. But the situation is not so trivial and I will try to show that the Quantum Chaos indeed exists in many facts: in the spectra of the energy levels, in remnants of the classical chaos, in the correlations and fluctuations of the spectra and so on.
I am not sure that the time will permit to touch the interesting connection of the Riemann hypothesis with the Quantum Chaos.
The talk is designed for non-specialists, but will also address some current topics.