Various recurrence and convergence results obtained in recent years indicate
that dynamical systems exhibit regular behavior along polynomial times.
While these results were mainly motivated by applications to
number theory and combinatorics, such as the polynomial extension of
Szemeredi's theorem on arithmetic progressions, this phenomenon also deserves attention
from the point of view of potential applications to physics. For example, the Poincare
recurrence theorem, as well as convergence theorems of the von Neumann and
Birkhoff type, hold along any sequence of the form p(n), n=1,2, ... where p(n) is a polynomial with integer coefficients satisfying p(0) = 0, and
it would be of interest
to give a physical interpretation of these facts. After reviewing some known results, we will
discuss the intriguing dichotomy between the theorems related to
polynomial and exponential behavior. The last part of the talk will be devoted to
open problems and conjectures.
Vitaly Bergelson studoval v Rusku a Izraeli. Jeho dizertace (vypracovana pod vedenim
prof. H. Furstenberga na Jeruzalemske univerzite) ziskala Landauovu cenu. Od roku
1984 je zamestnan na University of Ohio v Columbusu,
kde je v soucasnosti radnym profesorem. Prof. Bergelson je mezinarodne uznavanym
pracovnikem v ergodicke teorii a jejich aplikacich, zvlaste v kombinatoricke teorii
cisel. Spolecne se svym studentem A. Leibmanem dokazal napriklad
zname polynomialni verze Van der Waerdenovy vety a Szemerediho vety.
Vitaly je znamym prednasejicim a oblibenym
ucitelem. V letosnim roce prednesl zvanou prednasku na Mezinarodnim kongresu matematiku
v Madridu a rovne� Mordellovu prednasku v Cambridgi. Jeho pra�ske kolokvium je venovano
hlavni oblasti jeho zajmu a je urceno siroke matematicke a informaticke verejnosti.