Bolyai kollégium matematikai szemináriumának következõ programja

Szeptember 24-én és 25-én Geoffrey Grimmett Cambridge-i professzor tart kétrészes elõadást du. 1/4 5 órai kezdettel. A szerdai elõadás kivételes, a továbbiakban a Bolyai kollégium matematikai szemináriumának kezdési idõpontja csütörtök 1/4 5 óra. Viszont ki kívánjuk használni ezt a lehetõséget, ezért iktattunk be egy külön elõadást.

Minden potenciális érdeklõdõ figyelmét fel szeretném hívni erre az elõadásra. Grimmett professzor nemcsak matematikusnak, hanem elõadónak is kiváló. Egyik fõ erõssége, hogy nemcsak azt magyarázza el, hogy valamilyen eredmény miért igaz, hanem azt is, hogy az miért érdekes, és külön hangsúlyozza a legfontosabb gondolatokat.

Geoffrey Grimmett (Cambridge University) szeptember 24-én (szerda) és szeptember 25-én (csütörtök) 1/4 5 órai kezdettel tartandó kétrészes elõadásának absztraktja:

From Ferromagnetism to Stochastic Geometry

A piece of iron is placed in a magnetic field, which is allowed to increase in strength to some maximum and then decrease to zero. If the temperature is sufficiently low, then the iron will retain some magnetisation. There is a critical temperature above which no magnetisation is retained. This is a fundamental example of what is called a phase transition. The theory of phase transitions incorporates ideas of physics, probability theory, and geometry. The relevant models are examples of what are known as `Gibbs states', after J.W. Gibbs who published a fundamental monograph in 1902.

We shall give a brief introduction to the theory of Gibbs states, with special emphasis on the Ising and Potts models for ferromagnetism. There is tremendous interest in such systems, both for their `static' properties of phase transition, and for their dynamical properties as time passes. They have beautiful and complex structure.

The percolation process is a famous model for a disordered system of `pipes' through which liquid may pass. It is rather surprising that the percolation, Ising, and Potts models may be drawn togther in the same framework under the name `random-cluster model'. The presence of residual magnetisation in an Ising/Potts system corresponds to the existence of an infinite connected region in the random-cluster model.

The random-cluster model is simple and attractive to describe, and has many intrinsic properties involving probability and geometry. The theory is far from complete, and we shall present various `elementary' conjectures which are accepted widely as being true.

The two talks on these topics will be linked and in sequence, but it should be possible to follow the second without having attended the first.

Végül megjegyzem az érdeklõdõknek, hogy Grimmett professzor szeptember 26-án pénteken 11 órakor a Matematikai Kutató Intézet statisztikus fizika szemináriumán fog elõadni. Ennek az elõadásnak az absztraktja:

Stochastic Pin-Ball: Random Walks in Random Labyrinths

There is a version of the game of bagatelle, a pre-cursor of pin-ball, in which a ball rebounds off obstacles placed on a planar table. In a stochastic version of this game, obstacles are placed at random in $R^d$, and a ball ricochets off them with perfect reflection. What can be said about the trajectory of the ball? Known also as a `Lorentz lattice gas', there is a famous version of this problem termed the `Ehrenfest wind--tree model'.

Central properties of the trajectory are attainable if the environment is enriched with a positive density of places where the ball behaves in the manner of a random walk. In particular, one may establish theorems of non-localisation, recurrence/transience, and a central limit theorem, under conditions on the environment.

(Joint work with Bezuidenhout, Menshikov, Volkov, initiated in Budapest in 1995.)