# Seminars & Events for Topology Seminar

##### Combinatorial tangle Floer homology

In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss the construction of this combinatorial invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3gives back a stabilized version of knot Floer homology.

##### Symplectic fillings and star surgery

**This is a joint Topology-Symplectic Geometry seminar. Please note special time and location.** Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. In fact, even without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology. These fillings can be used in cut-and-paste operations called star surgery to construct examples of exotic 4-manifolds.

##### Dynamics and polynomial invariants of free-by-cyclic groups

The theory developed by Thurston, Fried and McMullen provides a near complete picture of the various ways a hyperbolic 3-manifold M can fiber over the circle. Namely, there are distinguished convex cones in the first cohomology H^1(M;R) whose integral points all correspond to fibrations of M, and the dynamical features of these fibrations are all encoded by McMullen's "Teichmuller polynomial." This talk will describe recent work developing aspects of this picture in the setting of a free-by-cyclic group G (the mapping torus group of an automorphism of a finite rank free group). Specifically, we will describe a polynomial invariant that determines a convex polygonal cone C in the first cohomology of G whose integral points all correspond to algebraically and dynamically interesting splittings of G.

##### Heegaard-Floer homology of algebraic links

The intersection of an algebraic plane curve singularity with a small 3-sphere is an algebraic link. I will explain how to compute the Heegaard-Floer homology of this link in terms of the homologies of certain subvarieties in the space of functions on the plane. The main technical tools are a spectral sequence computing Heegaard-Floer homology of L-space links and Orlik-Solomon theory of hyperplane arrangements. The talk is based on joint works 1301.7636 and 1403.3143 with Andras Nemethi.

##### A Morse-Bott approach to the Triangulation Conjecture

Manolescu has recently given a negative answer to the celebrated Triangulation Conjecture. His disproof relies on the construction of a new invariant of rational homology three spheres equipped with a spin structure. This is obtained by studying the Seiberg-Witten equations from the point of view of Conley index theory. In the present talk we discuss how to construct the analogous invariants in the Morse-theoretic framework of Kronheimer and Mrowka's monopole Floer homology. This approach works on every three manifold and is functorial under cobordisms.

##### All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups

By the inspirational works of Peter Sarnak and Ralph Phillips (ACTA 1985), we know that all classical Schottky groups (dim n >3) must have Hausdorff dimension strictly bounded away from dim n-1. Later Peter Doyle (ACTA 1988) showed that it is also true for n=3. But the natural question of nonclassical Schottky groups should have Hausdorff dimension strictly bounded from below away from 0 remains open. In this second part of our works on geometric structure of Kleinian groups of small Hausdorff dimensions we provide positive solution to this question. In particular we prove that there exists a universal positive number $\lambda>0$, such that any finitely-generated non-elementary Kleinian groups with limit set of Hausdorff dimension $<\lambda$ are classical Schottky groups.

##### 3-manifolds, Lipschitz geometry, and equisingularity

The local topology of isolated complex surface singularites is long understood, as cones on closed 3-manifolds obtained by negative definite plumbing. On the other hand a full understanding of the analytic types is out of reach, motivating Zariski's efforts into the 1980's to give a good concept of "equisingularity" for families of singularities. The significance of Lipschitz geometry as a tool in singularity theory is a recent insight, starting (in complex dimension 2) with examples of Birbrair and Fernandes published in 2008. I will describe work with Birbrair and Pichon on classifying this geometry in terms of discrete data associated with a refined JSJ decomposition of the associated 3-manifold link.

##### Taut foliations, left-orderability, and cyclic branched covers

##### Heegaard Floer homologies and cuspidal curves

##### Augmentation and sheaf categories for Legendrian knots

Recently Vivek Shende, David Treumann, and Eric Zaslow introduced a category of constructible sheaves associated to a Legendrian knot, and conjectured that it is equivalent to a category built from augmentations of Legendrian contact homology. In joint work in progress with Dan Rutherford, Steven Sivek, Shende, and Zaslow, it now appears that we can prove this conjecture. I'll describe both categories, with emphasis on the definition and properties of the augmentation category, and perhaps gesture toward the proof of their equivalence.

##### Contact structures and reducible surgeries

##### Knots and links from Thompson's groups

We will describe how to obtain all knots and links, oriented or not, as the coefficient of a certain infinite dimensional representation of Thompson's group F. The procedure also works for the group $T$.

##### Functoriality in Khovanov-Floer theories

There has been a lot of interest in recent years in connections between Khovanov homology and Floer theory. These connections usually come in the form of spectral sequences, with E_2 page the Khovanov homology of a link and converging to the relevant Floer theory. Important examples include Ozsvath-Szabo’s spectral sequence in Heegaard Floer homology and Kronheimer-Mrowka’s spectral sequence in singular instanton Floer homology. In particular, the latter was used to prove that Khovanov homology detects the unknot. A natural question is whether these constructions are functorial? That is, are the intermediate pages of these spectral sequences link invariants, and do link cobordisms induce well-defined maps on these pages? We answer these questions in the affirmative, as part of a much more general framework.

##### Burnside category and Khovanov homotopy type

I will review Khovanov homology in terms of a functor from the cube category to the category of (graded) abelian groups. I will describe how to factor this functor through the Burnside category, and how such a factorization allows us to refine Khovanov homology to a Khovanov stable homotopy type. This work is joint with Tyler Lawson and Robert Lipshitz.

##### Invariance of some spectral sequence from equivariant Floer homology

Recently, equivariant Floer homology has been used to construct a number of spectral sequences between Floer-type invariants of 3-manifolds and knots. We will discuss an alternate formulation of equivariant Lagrangian intersection Floer homology, with applications to invariance and computability of these spectral sequences. This is joint work in progress with Kristen Hendricks and Sucharit Sarkar, and is partly inspired by work of Paul Seidel's and a question of Tye Lidman’s.

##### Botany of transverse knots

We study transverse representatives of an oriented topological knot type K in the tight contact 3-sphere, where the classical invariant of transverse isotopy introduced by Bennequin is the self-linking number. The botany problem for transverse knots, namely deciphering which transverse isotopy classes exist at a fixed self-linking value, has remained open in general, although for certain knot types a number of interesting botanical features have been shown to exist. In this talk we present partial solutions of the botany problem that hold for arbitrary knot types K.

##### Instantons and odd Khovanov homology

I will describe a spectral sequence that starts at reduced odd Khovanov homology and converges to a version of instanton homology for double branched covers.

##### A discrete uniformization theorem for polyhedral surfaces

The classical uniformization theorem states that every Riemann surface carries a complete constant curvature Riemannian metric in its conformal class. It is difficult to algorithmically implement the uniformization theorem for polyhedral surfaces. We introduce a notion of discrete conformality for polyhedral surfaces and prove a discrete version of the uniformization theorem for compact polyhedral surfaces. This is a joint work with David Gu, Jian Sun and Tianqi Wu.

##### L-space slopes and bordered Heegaard Floer homology

It is conjectured that for irreducible 3-manifolds, being a non-L-space is equivalent both to having left-orderable fundamental group and to admitting a taut foliation. The latter two properties can be understood in some cases through cut and paste arguments, using a notion of detected slopes for 3-manifolds with toroidal boundary; we seek a similar understanding for L-spaces. For sufficiently nice 3-manifolds with torus boundary, I will describe how L-space slopes can be detected via bordered Heegaard Floer invariants, and I will determine when gluing two such manifolds produces an L-space. Our results, paired with work of Boyer and Clay, imply that the left-orderable/taut foliation/non L-space conjecture is true for graph manifolds with a single JSJ torus. This is joint work with Liam Watson.

##### The Cohomology of G-spaces (G compact group)

**Please note special time. **