Georgia Institute of Technology College of Sciences

The mapping class group is the group of symmetries of a surface (modulo homotopy). One way to study the mapping class group of a surface S is to understand its action on the set of simple closed curves in S (up to homotopy). The set of homotopy classes of simple closed curves can be organized into a simplicial complex called the complex of curves. This complex has some amazing features, and we will use it to prove a variety of theorems about the mapping class group. We will also state some open questions.

Knot contact homology (KCH) is a combinatorially defined topological invariant of smooth knots introduced by Ng. Work of Ekholm, Etnyre, Ng and Sullivan shows that KCH is the contact homology of the unit conormal lift of the knot. In this talk we describe a monodromy result for knot contact homology,namely that associated to a path of knots there is a connecting homomorphism which is invariant under homotopy. The proof of this result suggests a conjectural interpretation for KCH via open strings, which we will describe.

The study of Legendrian and transversal knots has been an essential part of contact topology for quite some time now, but until recently their study in overtwisted contact structures has been virtually ignored. In the past few years that has changed. I will review what is know about such knots and discuss recent work on the "geography" and "botany" problem.

We introduce two related sets of topological objects in the 3-sphere, namely a set of two-component exchangable links termed "iterated doubling pairs", and a see of associated branched surfaces called "Matsuda branched surfaces". Together these two sets possess a rich internal structure, and allow us to present two theorems that provide a new characterization of topological isotopy of braids, as well as a new characterization of transversal isotopy of braids in the 3-sphere endowed with the standard contact structure.

I will describe some results concerning factorizations ofdiffeomorphisms of compact surfaces with boundary. In particular, Iwill describe a refinement of the well-known \emph{right-veering}property, and discuss some applications to the problem ofcharacterization of geometric properties of contact structures interms of monodromies of supporting open book decompositions.

We will give definitions and then review a result by Floyd and Oertel that in a Haken 3-manifold M, there are a finite number of branched surfaces whose fibered neighborhoods contain all the incompressible, boundary-incompressible surfaces in M, up to isotopy. A corollary of this is that the set of boundary slopes of a knot K in S^3 is finite.

I will discuss moduli spaces of Riemannian metrics with various curvature conditions, and then focus on the case of nonnegative sectional curvature.

I will review results on the structure of open nonnegatively curved manifolds due to Cheeger-Gromoll, Perelman, and Wilking.

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham; The Georgia Institute of Technology; Emory University; The University of Tennessee Knoxville. The presentations will include topics on geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology.

Khovanov homology is an invariant of oriented links, that is defined as the cohomology of a chain complex built from the cube of resolutions of a link diagram. Discovered in the late 90s, it is the first of, and inspiration for, a series of "categorifications" of knot invariants. In this first of two one-hour talks, I'll give some background on categorification and the Jones polynomial, defineKhovanov homology, work through some examples, and give a portion of the proof of Reidemeister invariance.