Georgia Institute of Technology College of Sciences

I will discuss a computation of the lower central series of the Torelli group as a symplectic module, which depends on some conjectures and was performed 15 years ago in unpublished joint work with Ezra Getzler. Renewed interest in this computation comes from recent work of Benson Farb on representation stability.

The slice-ribbon conjecture states that a knot in $S^3=partial D^4$ is the boundary of an embedded disc in $D^4$ if and only if it bounds a disc in $S^3$ which has only ribbon singularities. In this seminar we will prove the conjecture for a family of Montesinos knots. The proof is based on Donaldson's diagonalization theorem for definite four manifolds.

I will define character varieties of finitely generated groups and consider some applications of them in geometry/topology.

Abstract: We utilize the Ozsvath-Szabo contact invariant to detect the action of involutions on certain homology spheres that are surgeries on symmetric links, generalizing a previous result of Akbulut and Durusoy. Potentially this may be useful to detect different smooth structures on $4$-manifolds by cork twisting operation. This is a joint work with S. Akbulut.

The theorem of Birman and Hilden relates the mapping class group of a surface and its image under a covering map. I'll explore when we can extend the original theorem and possible applications for further work.

I will consider two constructions which lead to information about the topology of a 3-manifold from one of its triangulation. The first construction is a modification of the Turaev-Viro invariant based on re-normalized 6j-symbols. These re-normalized 6j-symbols satisfy tetrahedral symmetries. The second construction is a generalization of Kashaev's invariant defined in his foundational paper where he first stated the volume conjecture. This generalization is based on symmetrizing 6j-symbols using *charges* developed by W. Neumann, S. Baseilhac, and R. Benedetti.