Schedule for: 16w5096 - Interactions of Gauge Theory with Contact and Symplectic Topology in Dimensions 3 and 4

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes (joint work with Mustafa Korkmaz) which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils on them.

In this talk, we discuss the character variety of traceless fundamental group representations to SU(2) of a tangle complement. Here, traceless means meridians of all n strands are sent to traceless SU(2) elements. There is a restriction map from the traceless character variety of an n-strand tangle to the traceless character variety of the 2n-punctured sphere. We’ll describe the structure of these traceless character varieties and their symplectic properties, and outline a program to define a Lagrangian Floer homology counterpart to Kronheimer Mrowka reduced singular instanton homology, which uses these traceless character varieties. This is joint work with Kirk and Hedden.

Transverse links in $S^3$ can be described via braids. We will show that the "direction and amount of twisting" of such a braid determine, in many cases, whether the (hat-version of) Heegaard Floer transverse invariant of the corresponding link vanishes or not. In particular, we prove that for 3-braids, the Heegaard Floer transverse invariant is non-zero if and only if the braid is right-veering. For higher-order braids, a fractional Dehn twist coefficient greater than 1 implies non-vanishing of the invariant. This result parallels a well-known result of Honda-Kazez-Matic for open books: if an open book with connected binding has FDTC > 1, then the Heegaard Floer contact invariant is non-zero. Interestingly, the open books result uses taut foliations and symplectic fillings (there is no direct proof) whereas our result for braids follows from the combinatorial structure of Dehornoy's braid orderings and an examination of grid diagrams.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Given a transverse knot in \(S^3\), together with an irrational number, one can define a filtration on the embedded contact homology of \(S^3\). This filtration is functorial with respect to symplectic cobordisms between transverse knots. Not much is known about this filtration in general (although it might be interesting to try to relate it to filtrations coming from Heegaard Floer theory). However the computation of this filtration for the unknot has significant implications for two-dimensional dynamics. Namely, for an area-preserving map of the disk, the filtration allows us to relate the mean action of periodic orbits to the Calabi invariant.

In this talk I will discuss the basic properties and examples of Pin(2)-monopole Floer homology (including some simple computational tools). This is the Morse-theoretic analogue of Manolescu's Pin(2)-equivariant Seiberg-Witten-Floer homology, and it can be used to provide an alternative disproof of the longstanding Triangulation Conjecture.

We survey some classical results about smooth foliations of three manifolds and then describe an approach, flow box neighborhoods, that has been useful in establishing related results for less smooth foliations that have been produced by many people, including Delman, Li, and Roberts. These results include \(C^0\) versions of the Eliashberg-Thurston approximation theorem, Calegari's leaf smoothing theorem, Dippolito's work on Denjoy blowups, and Tischler's fibration approximation theorem.

We survey some classical results about smooth foliations of three manifolds and then describe an approach, flow box neighborhoods, that has been useful in establishing related results for less smooth foliations that have been produced by many people, including Delman, Li, and Roberts. These results include \(C^0\) versions of the Eliashberg-Thurston approximation theorem, Calegari's leaf smoothing theorem, Dippolito's work on Denjoy blowups, and Tischler's fibration approximation theorem.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe joint work in progress with Tony Licata aimed at understanding an annular version of Lee's deformation of the Khovanov complex. In particular, we obtain a family of real-valued braid conjugacy class invariants generalizing Rasmussen's "s" invariant that give bounds on the Euler characteristic of smoothly-imbedded surfaces in the thickened solid torus. The algebraic model for this construction is the recently-defined Upsilon invariant of Ozsvath-Stipsicz-Szabo.

In a cooriented contact manifold, a positive Legendrian isotopy is a Legendrian isotopy evolving in the positive transverse direction to the contact plane. Their global behavior differs from the one of Legendrian isotopy and is closer to the one of propagating waves. In this talk I will explain how to use information in the Floer complex associated to a pair of Lagrangian cobordisms (recently constructed in a collaboration with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko) to give obstructions to certain positive loops of some Legendrian submanifolds. This will recover previously known obstructions and exhibit more examples. This is work in progress with V. Colin and G. Dimitroglou Rizell.

A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving several hundred thousand hyperbolic 3-manifolds, including a new practical construction of taut foliations on 3-manifolds via something I call a "laminar triangulation".

Bordered Floer homology is a variant of Heegaard Floer homology adapted to manifolds with boundary. I will describe a class of three-manifolds with torus boundary for which these invariants may be recast in terms of immersed curves in a punctured torus. This makes it possible to recast the paring theorem in bordered Floer homology in terms of intersection between curves leading, in turn, to some new observations about Heegaard Floer homology. This is joint work with Jonathan Hanselman and Jake Rasmussen.

I will explain how the ideas in the previous talk can be used to give simplified proofs of some earlier results due to Watson, Hanselman, Rasmussen, and Rasmussen. These theorems describe the set of L-space fillings of a manifold in terms of its Turaev torsion and characterize (under certain hypotheses) when a manifold containing an incompressible torus is an L-space. The orginal proofs were somewhat complicated, but in this new formalism everything is much simpler and more geometrical. This is joint work with Jonathan Hanselman and Liam Watson.

The talk concerns the singular Floer homology of knots and links defined by Kronheimer and Mrowka using gauge theory on orbifolds; it has been instrumental in proving that the Khovanov homology is an unknot detector. We show how replacing gauge theory on an orbifold with equivariant gauge theory on its double branched cover simplifies matters and allows for explicit calculations for several classes of knots and links. This is a joint work with Prayat Poudel.

We give new constraints on the topology of symplectic four-manifolds using invariants from Heegaard Floer homology. In particular, we will prove that certain simply-connected four-manifolds with positive-definite intersection forms cannot admit symplectic structures. This is joint work with Tye Lidman.

I will discuss a non-diagrammatic characterization of the class of alternating knots. I will focus on its 4-dimensional aspects and its application to the symmetries and crossing numbers of alternating knots.

I will describe the augmentation category, an A-infinity category associated to Legendrian submanifolds equipped with augmentations, and explain how one can use it to construct a derived Fukaya category for particular contact manifolds (1-jet spaces). The derived Fukaya category is generated by unknots, with the corollary that all augmentations "are geometric''. This is work in rather early progress with Tobias Ekholm and Vivek Shende, building on joint work with Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow, which itself builds on work of Frederic Bourgeois and Baptiste Chantraine.

We construct a Heegaard-Floer type homology from an open book decomposition supporting a contact structure. We will show that a special element, the contact class, vanishes when the contact structure is overtwisted and is non zero when it is fillable. We will further discuss invariance properties and, if time permits, relations with Symplectic Khovanov homology of a link. This is joint work with Ko Honda.

A well-known theorem of Dostoglou-Salamon establishes an Atiyah-Floer-type diffeomorphism between certain instanton moduli spaces and certain holomorphic curves moduli spaces. We give a new proof of this result that replaces much of the analysis with fairly straight-forward complex-geometric arguments.

I will define a version of Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds and discuss some topological applications.

In recent work, Baldwin and I defined invariants of contact 3-manifolds with boundary in sutured instanton Floer homology. I will sketch the proof of a theorem about these invariants which is analogous to a result of Plamenevskaya in Heegaard Floer homology: if a 4-manifold admits several Stein structures with distinct Chern classes, then the invariants of the induced contact structures on its boundary are linearly independent. As a corollary, we conclude that if a homology sphere Y admits a Stein filling with nonzero first Chern class, then there is a nontrivial representation (\pi_1(Y) \to SU(2)\). This is joint work in progress with John Baldwin.

I'll describe a simple proof (joint with Vela-Vick) that the rank of knot Floer homology detects the trefoil and that L-space knots are prime, results which were originally proven by Hedden-Watson and Krcatovich, respectively. Our argument is very Heegaard-diagram centric, but I'll describe an alternative proof which is more contact-geometric and uses Etnyre-Vela-Vick's "limit" description of knot Floer homology. The advantage of this geometric approach is that it can be (we think) ported to the instanton Floer setting to show that the rank of (sutured) instanton knot Floer homology detects the trefoil. If this all works it would, in combination with Kronheimer-Mrowka's spectral sequence relating Khovanov homology and singular instanton knot homology, prove that Khovanov homology detects the trefoil. The latter work is joint with Sivek.

To a group with two generators and one relator we assign a marked polytope and we show that if the group is the fundamental group of a 3-manifold, then the marked polytope determines the Thurston norm and the fibered cones of the 3-manifolds. This is joint work with Kevin Schreve and Stephan Tillmann.

One of active research areas in symplectic 4-manifolds is to cassify symplectic fillings of certain 3-manifolds equipped with a contact structure. Among them, people have long studied symplectic fillings of the link of a normal complex surface singularity. Note that the link of a normal complex surface singularity carries a canonical contact structure which is also known as the Milnor Fillable contact structure. One the other hand, algebraic geometers also have studied Milnor fibers as a general fiber of smoothings for a normal complex surface singularity. In this talk, I'd like to explain a relation between minimal symplectic fillings and the Milnor fibers of quotient surface singularities. Furthermore, if a time allows, I'd also like to investigate the relation for weighted homogeneous surface singularities. This is a joint work with Heesang Park, Dongsoo Shin, and Giancarlo Urz\'ua.

We outline a framework for defining a version of algebraic torsion for the contact element in Heegaard Floer Homology. This is work with Cagatay Kutluhan, Gordana Matic, and Andy Wand.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.