Georgia Institute of Technology College of Sciences

This is the first in the series of two talks aimed to discuss a recent work of Ontaneda which gives a poweful method of producing negatively curved manifolds. Ontaneda's work adds a lot of weight to the often quoted Gromov's prediction that in a sense most manifolds (of any dimension) are negatively curved.

To every homeomorphism of a surface, we can attach a positive real number, the entropy. We are interested in the question of what these homeomorphisms look like when the entropy is positive, but small. We give several perspectives on this problem, considering it from the complex analytic, surface topological, 3-manifold theoretical, and numerical points of view. This is joint work with Benson Farb and Chris Leininger.

A contact manifold with boundary naturally gives rise to a sutured manifold, as defined by Gabai. Honda, Kazez and Matic have used this relationship to define an invariant of contact manifolds with boundary in sutured Floer homology, a Heegaard-Floer-type invariant of sutured manifolds developed by Juhasz. More recently, Kronheimer and Mrowka have defined an invariant of sutured manifolds in the setting of monopole Floer homology. In this talk, I'll describe work-in-progress to define an invariant of contact manifolds with boundary in their sutured monopole theory.

In this talk I define the braid groups, its Garside structure, and its application to solve the word and conjugacy problems. I present a braid group with $n$ strands as the mapping class group of the disk with $n$ punctures, $\mathbb{D}^2-\{p_1\ldots p_n\}$, and a classification of surface homeomorphisms by the Nielsen Thurston theorem. I will also discuss results that require algebraic and geometric tools.

The main purpose of this talk is to better understand how to use branched covers to construct 3-manifolds. We will start with branched covers of 2-manifolds, carefully working through examples and learning the technology. Using these methods in combination with open book decompositions we will show how to construct 3-manifolds by branching over link and knots in S^{3}. Particular emphasis will be placed on using the map to get a "coloring" of the branched locus and how this combinatorial data is useful both for explicit constructions and for the general theory.

We will use a new concordance invariant, epsilon, associated to the knot Floer complex, to define a smooth concordance homomorphism. Applications include a new infinite family of smoothly independent topologically slice knots, bounds on the concordance genus, and information about tau of satellites. We will also discuss various algebraic properties of this construction.

I will discuss the Thurston norm for fibered hyperbolic 3-manifolds.

We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V>5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow, is based on a corresponding inequality for the numbers of double points, singularites, and inflections of closed contractible curves in the real projective plane which intersect every closed geodesic.

In this talk we will outline proof due to Plameneveskaya and Van-Horn Morris that every virtually overtwisted contact structure on L(p,1) has a unique Stein filling. We will give a much simplified proof of this result. In addition, we will talk about classifying Stein fillings of ($L(p,q), \xi_{std})$ using only mapping class group basics.

Continuation of last week's talk