Georgia Institute of Technology College of Sciences

Heegaard Floer theory provides a powerful suite of tools for studying 3-manifolds and their subspaces. In 2006, Ozsvath, Szabo and Thurston defined an invariant of transverse knots which takes values in a combinatorial version of this theory for knots in the 3—sphere. In this talk, we discuss a refinement of their combinatorial invariant via branched covers and discuss some of its properties. This is joint work with Mike Wong.

The talk will include a crash course on infinite dimensional topology, with applications to various topological properties of the space of congruence classes of convex bodies in the Euclidean space.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we sstart discussing branched covers of 3-manifolds.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we should be able to finish our discussion of branched covers of surfaces and transition to 3-manifolds.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we will continue studying branched covers of surfaces. Among other things we should be able to see how to use branched covers to see some relations in the mapping class group of surfaces.

The immersed Seifert genus of a knot $K$ in $S^3$ can be defined as the minimal genus of an orientable immersed surface $F$ with $\partial F = K$. By a result of Gabai, this value is always equal to the (embedded) Seifert genus of $K$. In this talk I will discuss the embedded and immersed cross-cap numbers of a knot, which are the non-orientable versions of these invariants. Unlike their orientable counterparts these values do not always coincide, and can in fact differ by an arbitrarily large amount.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). Along the way several open problems will be discussed.

Taffy pullers are machines designed to stretch taffy. They can modeled by surface homeomorphisms, therefore they can be studied by geometry and topology. I will talk about how efficiency of taffy pullers can be defined mathematically and what some of the open questions are. I will also talk about Macaw, a computer program I am working on, which does related computations and which will hopefully help answer some of the open questions.

Let S be an (n-1)-sphere smoothly embedded in a closed, orientable, smooth n-manifold M, and let the embedding be null-homotopic. We'll prove in the talk that, if S does not bound a ball, then M is a rational homology sphere, the fundamental group of both components of M\S are finite, and at least one of them is trivial. This talk is based on a paper of Daniel Ruberman.

Let M be a closed hyperbolic 3-manifold with a fibered face \sigma of the unit ball of the Thurston norm on H_2(M). If M satisfies a certain condition related to Agol’s veering triangulations, we construct a taut branched surface in M spanning \sigma. This partially answers a 1986 question of Oertel, and extends an earlier partial answer due to Mosher. I will not assume knowledge of the Thurston norm, branched surfaces, or veering triangulations.