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Series: Geometry Topology Seminar

Existence of a tight contact structure on a closed oriented three manifold is still widely open problem. In this talk we will present some work in progress to answer this problem for manifolds that are obtained by Dehn surgery on a knot in three sphere. Our method involves on one side generalizing certain geometric methods due to Baldwin, on the other unfolds certain homological algebra methods due to Ozsvath and Szabo.

Series: Geometry Topology Seminar

In the 1980’s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In turn, using the fact that the 2-fold cover of S^3 branched over the Whitehead double of a positive torus knot is negatively cobordant to a Seifert fibred homology sphere, Hedden-Kirk establish conditions under which an infinite family of Whitehead doubles of positive torus knots are independent in the smooth concordance group. In the talk, I will review some of the definitions and constructions involved in the proof by Hedden and Kirk and I will introduce some topological constructions that greatly simplify their argument. Time permiting I will mention some ways in which the result could be generalized to include a larger set of knots.

Series: Geometry Topology Seminar

In 2007, Honda, Kazez, and Matic defined an invariant of contact 3-manifolds with convex boundaries using sutured Heegaard Floer homology (SHF). Last year, Steven Sivek and I defined an analogous contact invariant using sutured Monopole Floer homology (SMF). In this talk, I will describe work with Sivek to prove that these two contact invariants are identified by an isomorphism relating the two sutured theories. This has several interesting consequences. First, it gives a proof of invariance for the contact invariant in SHF which does not rely on the relative Giroux correspondence between contact structures and open books (something whose proof has not yet been written down in full). Second, it gives a proof that the combinatorially computable invariants of Legendrian knots in Heegaard Floer homology can obstruct Lagrangian concordance.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Most work on surgeries in contact manifolds has focused upon determining the situations where tightness is preserved. We will discuss an approach to this problem from the reverse angle: when negative surgery on a fibred knot in an overtwisted contact manifold produces a tight one. We will examine the various phenomena that occur, and discuss an approach to characterising them via Heegaard Floer homology.

Series: Geometry Topology Seminar

We will start by counting lattice points in a polytope and showhow this produces many familiar objects in mathematics.For example if one scales the polytope, the number of lattice points givesrise to the Ehrhart polynomials, including binomals and other well knownfunctions.Things get more interesting once we take a weighted sum over the latticepoints instead of just counting them. I will explain how toextend Ehrhart's theory in this case and discuss an application to knottheory. We will derive a new state sum for the colored HOMFLYpolynomial using q-Ehrhart polynomials, following my recent preprint Arxiv1501.00123.

Series: Geometry Topology Seminar

How is the homological torsion of a hyperbolic 3-manifold related to its geometry? In this talk, I will explain some techniques to address this general question. In particular, I will discuss in detail the case of arithmetic manifolds, where the situation is presumably easier to understand.

Series: Geometry Topology Seminar

In this talk we will begin by discussing the problem of understanding the topology of the space of Riemannian metrics of positive scalar curvature on a smooth manifold. Recently much progress has occurred in this topic. We will then look at an application of the theory of operads to this problem in the case when the underlying manifold is an n-sphere. In the case when n>2, this space is a homotopy commutative, homotopy associative H-space. In particular, we show that it admits an action of the little n-disks operad. Via theorems of Stasheff, Boardman, Vogt and May, this allows us to demonstrate that the path component of this space containing the round metric, is weakly homotopy equivalent to an n-fold loop space.

Series: Geometry Topology Seminar

I will discuss Eliashberg and Thurston's theorem that C^2 taut foliations can be approximated by tight contact structures. I will try to explain the importance of their work and why it is useful to weaken their smoothness assumption. This work is joint with Rachel Roberts.

Series: Geometry Topology Seminar

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.