Georgia Institute of Technology College of Sciences

Defined in the early 2000's by Ozsvath and Szabo, Heegaard Floer homology is a package of invariants for three-manifolds, as well as knots inside of them. In this talk, we will describe how work from Poul Heegaard's 1898 PhD thesis, namely the idea of a Heegaard splitting, relates to the definition of this invariant. We will also provide examples of the kinds of questions that Heegaard Floer homology can answer. These ideas will be the subject of the topics course that I am teaching in Fall 2017.

Continuing from last time, we will discuss Hilden and Montesinos' result that every smooth closed oriented three manifold is a three fold branched cover over the three sphere, and also there is a representation by bands.

Let S be a Riemann surface of type (p,1), p > 1. Let f be a point-pushing pseudo-Anosov map of S. Let t(f) denote the translation length of f on the curve complex for S. According to Masur-Minsky, t(f) has a uniform positive lower bound c_p that only depends on the genus p.Let F be the subgroup of the mapping class group of S consisting of point-pushing mapping classes. Denote by L(F) the infimum of t(f) for f in F pseudo-Anosov. We know that L(F) is it least c_p.

In this series of talks we will show that every closed oriented three manifold is a branched cover over the three sphere, with some additional properties. In the first talk we will discuss some examples of branched coverings of surfaces and three manifolds, and a classical result of Alexander, which states that any closed oriented combinatorial manifold is always a branched cover over the sphere.

In this talk we associate a combinatorial dg-algebra to a cubic planar graph. This algebra is defined by counting binary sequences, which we introduce, and we shall provide explicit computations. From there, we study the Legendrian surfaces behind these combinatorial constructions, including Legendrian surgeries and the count of Morse flow trees, and discuss the proof of the correspondence between augmentations and constructible sheaves for this class of Legendrians.

I will describe a diagrammatic classification of (1,1) knots in S^3 and lens spaces that admit non-trivial L-space surgeries. A corollary of the classification is that 1-bridge braids in these manifolds admit non-trivial L-space surgeries. This is joint work with Sam Lewallen and Faramarz Vafaee.

Following up on the previous series of talks we will show how to construct Lagrangian Floer homology and discuss it properties.

Much of what is known about automorphisms of free groups is given by analogy to results on mapping class groups. One desirable result is the celebrated Nielson-Thurston classification of the mapping class group into reducible, periodic, or pseudo Anosov homeomorphisms. We will discuss attempts at analogous results for free group automorphisms.

Following up on the previous series of talks we will show how to construct Lagrangian Floer homology and discuss it properties.