My research in 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 certain tiling systems and as a consequence of this the undecidability, NP-completeness etc. problems have to be taken seriously. Basic questions concern the structure of the ensemble space, its size (topological entropy), existence of invariant measures and measures of maximal entropy. Through the last strong connections to Statistical Mechanics come around.
Recent activity has focused on finite systems where the long-range effects of the boundary of the domain drastically influences the geometry and statistics of the configuration in the interior (e.g. the Arctic Circle phenomenon). Such systems often admit a height representation and include Ice and Dimer models on various lattice as well as more general polyomino tilings.
Another interesting direction is the high 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 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 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