My research is within Dynamical Systems, Probability and Statistical Mechanics. It concentrates mostly on Symbolic and Measurable Dynamics, more specifically on subjects like
Multidimensional Markov-shiftsThese are generalizations of the subshifts of finite type to higher dimensions (the sequences are defined through finite block exclusion hence the Markovianity). One-dimensional symbolic dynamics with finite range interaction has a well developed theory but the higher dimensional case is considerably more subtle and its theory is still unfolding. The systems have close relations to tiling systems and as a consequence of this the undecidability, NP-completeness etc. problems have to be taken seriously. In the complement of these hard boundaries the accessible questions concern e.g. the structure of the ensemble space, its size (topological entropy), existence of invariant measures and measures of maximal entropy, in particular their multiplicity. Through the last (existence of phase transitions) strong connections to Statistical Mechanics come around.
Substantial activity focuses also on the finite versions systems where the long-range effects of the boundary of the domain can drastically influence the geometry and statistics of the configurations in the interior (the Arctic Circle phenomenon). Dimer, Ice, many more general Vertex models (on square lattice and models on non-standard lattices) as well as more general polyomino tilings admit height representation which is a key component in the analysis.
Another interesting direction is the high occupation density limits of some or these these models. In particular the so called hard core models yield striking insight into the nature of the densest packings in various ambient geometries.
Cellular automataare a special case of multidimensional Markov-shifts but because of their importance in physics and elsewhere in natural sciences their study warrants an extra set of questions to be asked. In them a uniform local rule governs the interaction dynamics of symbols distributed on a regular lattice (they are interacting particle systems if you like). In their purely deterministic form they are an excellent testing ground for the notions of dynamics (stability, attractor, generic behavior etc.) when infinite dimensional extensions are developed. In cellular automata context one can prove strong pseudorandomness results which explain why deterministic systems can exhibit highly random-like behavior. This complements the study of Lyapunov exponents and "chaos" in the context of smooth dynamcis.
For more details see the Publications page.
"Complete disorder is impossible." T.S. Motzkin