Thursday, September 13, 2018
09/13/2018 - 4:00pm
We show that every 3-uniform hypergraph with every vertex being contained in at least 5/9+o(1) proportion of all possible edges contains a (tight) Hamiltonian cycle. Several different lower bound constructions show that this degree condition is asymptotically optimal.
09/13/2018 - 4:15pm
The volume of a hyperbolic 3-manifold can be thought of as a measure of the “complexity” of the manifold. For hyperbolic manifolds of infinite volume instead of measuring the volume of the entire manifold one can instead take the volume of the “convex core” of the manifold which is typically finite. For infinite volume hyperbolic 3-manifolds a single topological manifold will support a large deformation space of hyperbolic structures and the convex core volume is an important function on this space. Unfortunately, this function isn’t smooth (for the natural differentiable structure on the deformation space). Krasnov-Schlenker have defined the notion of “renormalized volume” motivated by work of Graham-Witten on conformally compact Einstein manifolds. The renormalized volume is coarsely the same as the convex core volume but has the advantage that is it a smooth function and further Krasnov-Schlenker have given a simple formula for its derivative. We will define this concept and then describe some of our joint work with J. Brock and M. Bridgeman studying its gradient flow.