Georgia Institute of Technology College of Sciences

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.

I will give an example of transforming a knot into closed braid form using Yamada-Vogel algorithm. From this we can write down the corresponding element of the knot in the braid group. Finally, the definition of a colored Jones polynomial is given using a Yang-Baxter operator. This is a preparation for next week's talk by Anh.

The talk will be about my ongoing work on spaces of complete non-negatively curved metrics on low-dimensional manifolds, such as Euclidean plane, 2-sphere, or their product.

In this talk, I will introduce a notion of geometric complexity  to study topological rigidity of manifolds. This is joint work with Erik Guentner and Romain Tessera. I will try to make this talk accessible to graduate students and non experts.

( This will be a continuation of last week's talk. )An n-dimensional topological quantum field theory is a functor from the category of closed, oriented (n-1)-manifolds and n-dimensional cobordisms to the category of vector spaces and linear maps. Three and four dimensional TQFTs can be difficult to describe, but provide interesting invariants of n-manifolds and are the subjects of ongoing research. This talk focuses on the simpler case n=2, where TQFTs turn out to be equivalent, as categories, to Frobenius algebras. I'll introduce the two

I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S^3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.