Research overview
My research is within Dynamical Systems, Probability and Statistical Mechanics. It has concentrated mostly on Symbolic and Measurable Dynamics, more specifically on subjects like
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.
An challenging class of systems here are deterministic cellular automata which in some special case admit even rigorous results. They are also 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 thereby complementing 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
Multidimensional Markov-shifts
These 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.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.
An challenging class of systems here are deterministic cellular automata which in some special case admit even rigorous results. They are also 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 thereby complementing the study of Lyapunov exponents and "chaos" in the context of smooth dynamcis.
Bounded Statistical Mechanics Models
Substantial activity focuses on the finite versions of various Statistical Mechanics 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 are included here. Equilibrium computation through relaxation of course ties the studies to more general Dynamics formulations.For more details see the Publications page.
"Complete disorder is impossible." T.S. Motzkin