Georgia Institute of Technology College of Sciences

In this work we provide several improvements in the study of phase transitions of interacting particle systems: 1. We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of `sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus in the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. 2. We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogue hard-hexagon in 1980. 3. We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the `clustering' threshold of such model; thus providing further evidence for the conjectural algorithmic `hardness' occurring at such point.

I will define character varieties of finitely generated groups and consider some applications of them in geometry/topology.

Given a non-null vector field X in a Riemannian manifold, a hypersurfaceis said to have a canonical principal direction relative to $X$ if theprojection of X onto the tangent space of the hypersurface gives aprincipal direction. We give different ways for building thesehypersurfaces, as well as a number of useful characterizations. Inparticular, we relate them with transnormal functions and eikonalequations. Finally, we impose the further condition of having constantmean curvature to characterize the canonical principal direction surfacesin Euclidean space as Delaunay surfaces.

In this talk a curve complex HC(S) closely related to the "Cyclic Cycle Complex" (Bestvina-Bux-Margalit) and the "Complex of Cycles" (Hatcher) is defined for an orientable surface of genus g at least 2. The main result is a simple algorithm for calculating distances and constructing quasi-geodesics in HC(S). Distances between two vertices in HC(S) are related to the "Seifert genus" of the corresponding link in S x R, and behave quite differently from distances in other curve complexes with regards to subsurface projections.