MARK KAC SEMINAR

On Z^2, the set of extremal states of the subcritical Ising model consists of the two pure plus and minus phases. On Z^3 is conjectured to be at most countable at low temperature. On a regular tree, it is known to be uncountable at low temperature. I will explain how to exhibit an uncountable family of extremal (non-automorphism invariant) Gibbs measures of the low temperature Ising model on non-amenable graphs. These states arise as low temperature perturbations of local groundstates having a sparse enough set of frustrated edges, the sparseness being measured in terms of the isoperimetric constant of the graph. On regular tilings of the hyperbolic plane, our techniques allow to deduce extremality of uncountably many Series-Sinai states (having one interface along a continuous geodesic of H^2), and lead to the study of nice questions about hyperbolic billiards. Based on joint works with Matteo D’Achille, Christof Kuelske, Arnaud Le Ny and Jean Vereecke.

Interacting particle models are an indispensible tool in multidisciplinary collaborations. In this talk, I will describe my work from several such application areas, introducing the interacting particle models I design and discussing any corresponding continuum models. In particular, I will begin withan off-lattice model for the migration of the Icelandic capelin and highlight open questions. I will transition to particle models for humans, where Ihave modeled gangs and stressed crowds using both lattice and off-lattice models. I will begin with a lattice-based model for gang territorial dynamics and formally derive a coupled system of PDEs which allows us toidentify the critical values for a phase transition between well-mixed and segregated territories. I will then discuss recent work on mathematical oncology. There, we employ similar techniques to model the spatial dynamics of treatment-resistant prostate cancer, formally deriving a set of coupled PDEsand using the model to explore alternative treatment strategies. I will end by discussing current experiments in the Cube, a motion-capture studio at Virginia Tech, where we have a multidisciplinary team of mathematicians, engineers, and social pyschologists working together to explore the effects of stress in evacutation dynamics. Along the way, I will mention open problems where I believe probability theory would be helpful.