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Series: Geometry Topology Seminar

This is joint work with Rob Kirby. Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds; a Heegaard splitting splits a 3-manifolds into 2 pieces each of which looks like a regular neighborhood of a bouquet of circles in R^3 (a handlebody), while a trisection splits a 4-manifold into 3 pieces of each of which looks like a regular neighborhood of a bouquet of circles in R^4. All closed, oriented 4-manifolds (resp. 3-manifolds) have trisections (resp. Heegaard splittings), and for a fixed manifold these are unique up to a natural stabilization operation. The striking parallels between the two dimensions suggest a plethora of interesting open questions, and I hope to present as many of these as I can.

Series: Geometry Topology Seminar

An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the story is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is “finitely many”? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on a quantitative version of the “how many?” question.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

We will define transverse surgery, and study its effects on open books, the Heegaard Floer contact invariant, and tightness. We show that surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than 2g-1. We also give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

Series: Geometry Topology Seminar

In this talk I will explain the Dynnikov’s coordinate system, which puts global coordinates on the boundary of Teichmuller space of the finitely punctured disk, and the update rules which describe the action of the Artin braid generators in terms of Dynnikov’s coordinates. If time permits, I will list some applications of this coordinate system. These applications include computing the geometric intersection number of two curves, computing the dilatation and moreover studying the dynamics of a given pseudo-Anosov braid on the finitely punctured disk.

Series: Geometry Topology Seminar

A well known result of Giroux tells us that isotopy classes ofcontact structures on a closed three manifold are in one to onecorrespondence with stabilization classes of open book decompositions ofthe manifold. We will introduce a characterization of tightness of acontact structure in terms of corresponding open book decompositions, andshow how this can be used to resolve the question of whether tightness ispreserved under Legendrian surgery.

Series: Geometry Topology Seminar

In this talk we will discuss an ODE associated to the evolution of curvature along the Ricci flow. We talk about the stability of certain fixed points of this ODE (up to a suitable normalization). These fixed points include curvature of a large class of symmetric spaces.

Series: Geometry Topology Seminar

In joint work with Joan Birman and Bill Menasco, we describe a new finite set of geodesics connecting two given vertices of the curve complex. As an application, we give an effective algorithm for distance in the curve complex.

Series: Geometry Topology Seminar

Heegaard Floer theory consists of a set of invariants of three-and four-dimensional manifolds. Three-manifolds with the simplest HeegaardFloer invariants are called L-spaces and the name stems from the fact thatlens spaces are L-spaces. The primary focus of this talk will be on thequestion of which knots in the three-sphere admit L-space surgeries. Wewill also discuss about possible characterizations of L-spaces that do notreference Heegaard Floer homology.

Series: Geometry Topology Seminar

A monoidal subset of a group is any set which is closed under the product (and contains the identity). The standard example is Dehn^+, the set of maps whcih can be written as a product of right-handed Dehn twists. Using open book decompositions, many properties of contact 3-manifolds are encoded as monoidal subsets of the mapping class group. By a related construction, contact topology also produces a several monoidal subsets of the braid group. These generalize the notion of positive braids and Rudolphs ideas of quasipositive and strongly quasipositive. We'll discuss the construction of these monoids and some of the many open questions.