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Series: Geometry Topology Seminar

Since its introduction in 1966 by Fox and Milnor the knot concordance group has been an invaluable algebraic tool for examining the relationships between 3- and 4- dimensional spaces. Though knots generalize naturally to links, this group does not generalize in a natural way to a link concordance group. In this talk, I will present joint work with Matthew Hedden where we define a link concordance group based on the “knotification” construction of Peter Ozsvath and Zoltan Szabo. This group is compatible with Heegaard Floer theory and, in fact, much of the work on Heegaard Floer theory for links has implied a study of these objects. Moreover, we have constructed a generalization of Milnor’s group-theoretic higher order linking numbers in a novel context with implications for our link concordance group.

Series: Geometry Topology Seminar

The study of transverse knots in dimension 3 has been instrumental in the development of 3 dimensional contact ge- ometry. One natural generalization of transverse knots to higher dimensions is contact submanifolds. Embeddings of one contact manifold into another satisfies an h-principle for codimension greater than 2, so we will discuss the case of codimension 2 contact embeddings. We will give the first pair of non-isotopic contact embeddings in all dimensions (that are formally isotopic).

Series: Geometry Topology Seminar

We show that for any connected sum of lens spaces L there exists a connected sum of lens spaces X such that X is rational homology cobordant to L and if Y is rational homology cobordant to X, then there is an injection from H_1(X; Z) to H_1(Y; Z). Moreover, as a connected sum of lens spaces, X is uniquely determined up to orientation preserving diffeomorphism. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

We will describe a geometric interpretation of Khovanov homology as Lagrangian Floer homology of two immersed curves in the 4-punctured 2-dimensional sphere. The main ingredient is a construction which associates an immersed curve to a 4-ended tangle. This curve is a geometric way to represent Khovanov (or Bar-Natan) invariant for a tangle. We will show that for a rational tangle the curve coincides with the representation variety of the tangle complement. The construction is inspired by a result of [Hedden, Herald, Hogancamp, Kirk], which embeds 4-ended reduced Khovanov arc algebra (or, equivalently, Bar-Natan dotted cobordism algebra) into the Fukaya category of the 4-punctured sphere. The main tool we will use is a category of peculiar modules, introduced by Zibrowius, which is a model for the Fukaya category of a 2-sphere with 4 discs removed. This is joint work with Claudius Zibrowius and Liam Watson.

Series: Geometry Topology Seminar

Unknotting number is one of the simplest, yet mysterious, knot invariants. For example, it is not known whether it is additive under connected sum or not. In this talk, we will construct lower bounds for the unknotting number using two homological knot invariants: knot Floer homology, and (variants of) Khovanov homology. Unlike most lower bounds for the unknotting number, these invariants are not lower bound for the slice genus and they only vanish for the unknot. Parallely, we will discuss connections between knot Floer homology and (variants of) Khovanov homology. One main conjecture relating knot Floer homology and Khovanov homology is that there is a spectral sequence from Khovanov homology to knot Floer homology. If time permits, we will sketch an algebraically defined knot invariant, for which there is a spectral sequence from Khovanov homology converging to it. The construction is inspired by counting holomorphic discs, so we expect it to recover the knot Floer homology. This talk is based on joint works with Eaman Eftekhary and Nathan Dowlin.

Series: Geometry Topology Seminar

Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g ≥ 2. Given a finite subgroup H < Mod(Sg), let Fix(H) denote the set of fixed points induced by the action of H on the Teichmuller space Teich(Sg). In this talk, we give an explicit description of Fix(H), when H is cyclic, thereby providing a complete solution to the Modular Nielsen Realization Problem for this case. Among other applications of these realizations, we derive an intriguing correlation between finite order maps and the filling systems of surfaces. Finally, we will briefly discuss some examples of realizations of two-generator finite abelian actions.

Series: Geometry Topology Seminar

This will be a 90 minute seminar

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically/Stein fillable when r is in (0,1]. This is joint work with James Conway and John Etnyre.

Series: Geometry Topology Seminar

The h-principle is a powerful tool in differential topology which is used to study spaces of functionswith certain distinguished properties (immersions, submersions, k-mersions, embeddings, free maps, etc.). Iwill discuss some examples of the h-principle and give a neat proof of a special case of the Smale-HirschTheorem, using the "removal of singularities" h-principle technique due to Eliashberg and Gromov. Finally, I willdefine and discuss totally convex immersions and discuss some h-principle statements in this context.

Series: Geometry Topology Seminar

Novikov homology was introduced by Novikov in the
early 1980s motivated by problems in hydrodynamics. The Novikov
inequalities in the Novikov homology theory
give lower bounds for the number of critical points of a Morse closed 1-form on a compact
differentiable manifold M. In the first part of my talk I shall
survey the Novikov homology theory in finite dimensional setting
and its further developments in infinite dimensional setting with applications in the theory of symplectic fixed points and Lagrangian intersection/embedding problems. In the second part of my
talk
I shall report on my recent joint work with Jean-Francois Barraud and Agnes Gadbled on construction of the Novikov fundamental group
associated to a cohomology
class of a closed 1-form on M and its application to obtaining
new lower bounds for the number of
critical points of a Morse 1-form.