# Seminars & Events for Topology Seminar

##### Comultiplication and the Ozsvath-Szabo contact invariant

Let $S$ be a surface with boundary and suppose that $g$ and $h$ are diffeomorphisms of $S$ which restrict to the identity on the boundary. I'll describe how the Ozsvath-Szabo contact invariants associated to the open books $(S,g)$, $(S,h)$, and $(S,hg)$ are natural with respect to a comultiplication on the corresponding Heegaard-Floer homology groups. In particular, it follows that if the contact invariants associated to the open books $(S,g)$ and $(S,h)$ are non-zero, then so is the contact invariant associated to the open book $(S,hg)$. I plan to discuss an extension of this comultiplication to ${HF}^+$ and an obstruction to the compatibility of a contact structure with a planar open book.

##### Components of Springer fibers and Khovanov's arc algebra

Using the structure of certain Springer fibers and their components, I'll describe a geometric construction of an algebra which is painfully close to being isomorphic to the arc algebra defined by Khovanov, but in fact, isn't. I'll then hopefully explain why this is actually a good thing.

##### Subdirect products of surfaces, homological finiteness, and residually-free groups

##### Deformation of a hyperbolic 4-orbifold

It is well known that Thurston's beautiful deformation theory of hyperbolic structures is mostly useless in dimensions > 3. Steve Kerckhoff and I have been studying a new example of a hyperbolic deformation in 4 dimensions which produces an infinite number of new hyperbolic 4-orbifolds with interesting properties. The talk will attempt to motivate this work. It will be aimed at a general geometry/topology audience.

##### Invariants of Legendrian knots in Heegaard Floer homology

A new invariant of Legendrian knots will be defined, taking values in the knot Floer homology of the underlying null-homologous knot. With the aid of this invariant we find transversely non-simple knots in many overtwisted contact structures, and show that the Eliashberg-Chekanov twist knots (in particular the $7_2$ knot in Rolfsen's table) are not transversely simple.

##### Real Projective Structures and Non-standard analysis

We investigate the analog of the Thurston boundary of Teichmüller space in the context of convex real projective structures on closed manifolds. In particular we give a new interpretation of measured laminations in terms of non-standard hyperbolic structures over the hyper-reals.

##### Variational principles on triangulated surfaces

We will discuss various applications of recently discovered 2-dimensional counterparts of the Schlaefli formula.

##### Sequences of Hyperbolic $3$-Manifolds with Unfaithful Markings

Let $\Gamma$ be a finitely generated group. To every representation $\rho : \Gamma\to Isom (BH3)$ with discrete and torsion-free image there corresponds a hyperbolic $3$-manifold $M_\rho = BH3 / \rho (\Gamma)$. I will present some new results linking the pointwise convergence of a sequence of such representations with Gromov-Hausdorff convergence of the corresponding quotient manifolds. A detailed analysis already exists for sequences of faithful representations; I will give examples that illustrate the failure of these theorems in the unfaithful setting, and offer some useful replacements. Joint work with Juan Souto.

##### Length Spectrum of a Flat Metric

I'll discuss joint work-in-progress with M. Duchin and K. Rafi on the geometry of flat structures on surfaces via the lengths of its closed geodesics.

##### On the renormalized volume of quasifuchsian manifolds

The renormalized volume of quasifuchsian hyperbolic 3-manifolds was originally introduced for physical reasons. Takhtajan and Zograf (and others) discovered that it provides a Kähler potential for the Weil-Petersson metric on Teichmüller space. We will give an elementary, differential-geometric account of this result. It can be extended to quasifuchsian manifolds having cone singularities along infinite lines, yielding results on the Teichmüller space of hyperbolic metrics with cone singularities (of prescribed angles) on a closed surface. (Based on joint works with K. Krasnov, C. Lecuire, S. Moroianu.)

##### Loop products and closed geodesics

The critical points of the energy function on the free loop space $L(M)$ of a compact Riemannian manifold $M$ are the closed geodesics on $M$. Filtration by the length function gives a link between the geometry of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of $L(M)$. Geometry reveals the existence of a related product on the cohomology of $L(M)$. For manifolds such as spheres and projective spaces for which there is a metric with all geodesics closed, the resulting homology and cohomology rings are nontrivial, and closely linked to the geometry. I will not assume any knowledge of the Chas-Sullivan product. Joint work with Mark Goresky.

##### The convexity of length functions on Fenchel-Neilsen coordinates for Teichmuller space

##### HF-hat for 3-manifolds with boundary, via a toy example

We will start by sketching the nature of our extension of HF-hat to 3-manifolds with boundary. Following this, we will focus on a toy model, in terms of planar grid diagrams, in which the main aspects of the theory can be readily seen. We will conclude by mentioning the additional complications not present in the toy model.

This is joint work with P. Ozsvath and D. Thurston.