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Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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series of the Torelli group as a symplectic module, which depends on some conjectures and was performed 15 years ago in unpublished joint work with Ezra Getzler.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.

Series: Geometry Topology Seminar

Given a non-null vector field X in a Riemannian manifold, a hypersurfaceis said to have a canonical principal direction relative to $X$ if theprojection of X onto the tangent space of the hypersurface gives aprincipal direction. We give different ways for building thesehypersurfaces, as well as a number of useful characterizations. Inparticular, we relate them with transnormal functions and eikonalequations. Finally, we impose the further condition of having constantmean curvature to characterize the canonical principal direction surfacesin Euclidean space as Delaunay surfaces.

Series: Geometry Topology Seminar

In this talk a curve complex HC(S) closely related to the "Cyclic Cycle Complex" (Bestvina-Bux-Margalit) and the "Complex of Cycles" (Hatcher) is defined for an orientable surface of genus g at least 2. The main result is a simple algorithm for calculating distances and constructing quasi-geodesics in HC(S). Distances between two vertices in HC(S) are related to the "Seifert genus" of the corresponding link in S x R, and behave quite differently from distances in other curve complexes with regards to subsurface projections.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

In this talk I will describe how the quantum determinant modelof the Colored Jones polynomial, developed by Vu Huynh and Thang Le can beinterpreted in a combinatorial way as walks along a braid. Thisinterpretation can then be used to prove that the leading coefficients ofthe colored Jones polynomial stabalize, defining two power series calledthe head and the tail. I will also show examples where the head and tailcan be calculated explicitly and have applications in number theory.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

The actual talk will be 40 minutes. Note the unusual time.

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.

Series: Geometry Topology Seminar

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. In this talk, I will focus on the example of nilpotent representations of quantized sl(2) at a root of unity. In this example, the two constructions are equal and give rise to a kind of Homotopy Quantum Field Theory. This is joint work with R. Kashaev, B. Patureau and V. Turaev.

Series: Geometry Topology Seminar

I will talk about some progress in proving the Degree Conjecture for torus
knots. The conjecture states that the degree of a colored Jones polynomial colored by an
irreducible representation of a simple Lie algebra g is locally a quadratic
quasi-polynomial. This is joint work with Stavros Garoufalidis.