Georgia Institute of Technology College of Sciences

The Weeks manifold W is a closed orientable hyperbolic 3-manifold with the smallest volume. Understanding contact structures on hyperbolic 3-manifolds is one of problems in contact topology. Stipsicz previously showed that there are 4 non-isotopic tight contact structures on the Weeks manifold. In this talk, we will exhibit 7 non-isotopic tight contact structures on W with non-vanishing Ozsvath-Szabo invariants.

On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the edges with common distribution F. For which F is there an infinite self-avoiding path with finite total weight? This question arises in first-passage percolation, the study of the random metric space Z^2 with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when F(0)<1/2 (there are only finite paths of zero-weight edges), and there is one when F(0)>1/2 (there is an infinite path of zero-weight edges). The critical case, F(0)=1/2, is considerably more difficult due to the presence of finite paths of zero-weight edges on all scales. I will discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient conditions on F for the existence of an infinite finite-weight path. The methods involve comparing the model to another one, invasion percolation, and showing that geodesics in first-passage percolation have the same first order travel time as optimal paths in an embedded invasion cluster.

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random particle simulations, flocks and mills, and their qualitative behavior. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. Compactly supported global minimizers determine the flock patterns whose existence is related to the classical H-stability in statistical mechanics and the classical obstacle problem for differential operators.