Background Talk

Title: An Introduction to Knot Floer Homology

Abstract: Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting holomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. I will sketch an outline of the original construction, and then discuss grid homology in greater detail. 

Research Talk

Title: GRID Invariants Generalizations and Lagrangian Cobordisms

Abstract: In 2006, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. I will describe various generalizations of these GRID invariants, and use them to obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure. Based on collaborations with: Jubeir, Schwartz, Winkeler, and Wong; Lewis, Ojakli, and Shapiro; Wong.

Background Talk

Title: Symplectic and contact toric manifolds

Abstract: I will first explain basic properties of toric actions on closed symplectic  manifolds. Then, I will introduce  toric actions on symplectic manifolds with a convex or concave contact type boundary. Such a toric action preserves the contact structure on the boundary, and, therefore, the boundary itself is a contact toric manifold. To every contact toric manifold we associate a moment cone, a moment map image of the symplectisation, that also admits a toric action. I will describe how to recognise from the moment cone if the contact toric 3-manifold is tight or overtwisted.

Research Talk

Title: Concave contact structures on linear plumbings and generalisations

Abstract: In this talk I  will first recall  the plumbing construction of 4-manifolds, according to the associated plumbing graphs. Plumbings are introduced in 60th's by Milnor and Wall, and later developed by Hirzebruch, Neumann etc. Then, I will explain how to impose a symplectic structure with a contact type boundary on these manifolds.   When all base surfaces are spheres and the plumbing is linear, I will show that, if at least one self-intersection number of base spheres is non-negative, the plumbing admits a structure of a symplectic toric manifold with a concave contact boundary.   I will explain how to determine if this contact structure is tight or overtwisted, by looking into the corresponding moment cone that depends only on self-intersection numbers of base spheres. I will extend these results to certain non-linear plumbings. This talk is based on a joint work with Jo Nelson, Ana Rechtman, Laura Starkston, Shira Tanny and Luya Wang.

Title: Calculating Spectral Order in Heegaard Floer Homology

Abstract: We describe an invariant of contact structures in dimension three that arises as a refinement of the Ozsvath-Szabo contact class in Heegaard Floer homology.  This "spectral order" invariant takes values in the set $Z_{\ge 0} \cup {\infty}$. It is zero for overtwisted contact structures, and computable (in a limit) from any supporting open book decomposition. It is nondecreasing under Legendrian surgery, hence ${\infty}$ for Stein-fillable contact structures, so finiteness of the invariant is an obstruction to Stein fillability. We give some examples and describe an algorithm to compute the spectral order associated to a given open book and a basis of arcs on the page. This is joint work with Cagatay Kutluhan, Jeremy Van Horn-Morris and Andy Wand.

Title: Ping-Pong and Beyond via Dynnikov Coordinates

Abstract: Dynnikov coordinates give a combinatorial description of curves on punctured disks, where mapping class group actions are expressed by piecewise-linear update rules. In this talk, I use these coordinates to study subgroups generated by relaxed curves. For configurations of relaxed curves with prescribed intersection properties, I define regions in the coordinate space and use them to implement a ping-pong argument for freeness. The update rules allow us to track how these regions behave under the action by following how inequalities between coordinate entries change.

The argument is based on the use of the update rules together with explicit computations. We also get a refinement of a classical intersection-type inequality, and a coordinate-based description of constructions of free and right-angled Artin subgroups. This work was carried out within the scope of a TÜBİTAK No:123F221 research project.

Background Talk

Title: On embedding problems for 3-manifolds in 4-space

Abstract: In this pre-talk I will focus on various 3-manifold embedding problems in 4-dimensional Euclidean space R^4. Given a closed, orientable 3-manifold Y, it is of great interest but often a difficult problem to determine whether Y may be smoothly embedded in R^4. This is the case even for integer homology spheres, and restricting to special classes such as Seifert manifolds; the problem is open in general. On the other hand, under additional geometric considerations coming from symplectic geometry (for example, hypersurfaces of contact type in R^4) or complex geometry (such as the boundaries of holomorphically or rationally convex domains in complex Euclidean space C^2), the problems often become tractable and in certain cases a uniform answer is possible. I will survey these perspectives, explain how they relate to one another and highlight related conjectures and open problems.

Research Talk

Title: On Weinstein domains in symplectic manifolds

Abstract: In this talk, I will focus will be on the topology of contact type hypersurfaces in symplectic manifolds. More specifically, I will investigate the question: given a (simple) symplectic manifold X, which manifolds can arise as contact type hypersurfaces in X? The situation in dimension 4, which will be our primary focus, is expected to be significantly more constrained. After providing some background and context, I will report on recent joint work with T. Mark that identifies an obstruction for a contact 3-manifold to embed in certain closed symplectic 4-manifolds as the boundary of a Weinstein domain -- a stronger requirement than contact type. I will also explore consequences of this obstruction for the symplectic topology of such domains in small rational surfaces.

Background Talk

Title: Mapping class group monoids in contact topology

Abstract: Many fundamental properties of contact structures can be characterized by monoids in surface mapping class groups; this viewpoint moreover provides a particularly accessible framework for introducing such properties.  This talk will describe a number of these monoids, focusing on background concepts and relevance to current research. 

Research Talk

Title: Positivity and Stein fillability of contact 3-manifolds

Abstract: We will discuss motivation for and approaches to the question of when the monoid in the mapping class group of a surface with boundary corresponding to monodromies of open book decompositions of Stein fillable contact 3-manifolds differs from the monoid of mapping classes which admit factorizations into positive Dehn twists. In particular, combining new(ish) results with previous work of several people, we give a complete solution to this problem, showing that the monoids coincide only for planar surfaces. This is joint work with Vitalijs Brejevs.

Title: Fast computations for pseudo-Anosov mapping classes 

Abstract: We present a quadratic-time algorithm for computing the stretch factor and invariant measured foliations of a pseudo-Anosov mapping class given as a word in the generators, in joint work with Margalit, Strenner, and Taylor. This gives the first sub-exponential time algorithm for the problem. We also introduce a tropical Laurent polynomial representation of the mapping class group of a punctured surface, developed with Alessandrini, via explicit formulae describing the action of each generator in terms of normal coordinates, which we use to illustrate the algorithm.