Schedule for: 20w5088 - Interactions of gauge theory with contact and symplectic topology in dimensions 3 and 4 (Online)

A brief introduction and welcome to BIRS Online.

I'll discuss ongoing joint work with Katherine Raoux that uses knot Floer homology to establish relative adjunction inequalities. These inequalities bound the Euler characteristics of properly embedded smooth cobordisms between links in the boundary of certain smooth 4-manifolds. The inequalities generalize the slice genus bound for the "tau" invariant studied by Ozsvath-Szabo and Rasmussen. I will use our inequalities to define concordance invariants of links, prove new results about contact structures, motivate a 4-dimensional interpretation of tightness, and to show that knots with simple Floer homology in lens spaces (or L-spaces) minimize rational slice genus amongst all curves in their homology class, upgrading a remarkable result of Ni and Wu pertaining to the rational Seifert genus.

We will use the Ozsváth-Szabó and Kronheimer-Mrowka spectral sequences to show that the module structure on Khovanov homology detects split links. This is joint work with Sucharit Sarkar.

We will have two breakout rooms for discussions after the talks.

The relationship between contact topology and various Floer homologies has been a fundamental tool to settle open question in low dimensional topology. The contact invariant in Heegaard Floer homology was one of the main instrument in these applications. In this talk I will extend the definition of the contact invariant for bordered Floer homology. The bordered contact invariant satisfies a gluing formula and recovers the contact invariant for closed and sutured manifolds. The main tools for this extension are foliated open books, and I will spend most of the time explaining these, and another application concerning the additivity of the support norm for tight contact structures. Parts of this talk is joint work with Alishahi, Foldvari, Hendricks, Licata, and Petkova.

When I was a graduate student, Dave Gay asked me if I could find a way to "see" when a 4-manifold X admits a symplectic structure via a trisection of X. I've finally got the answer, but it took a surprising route to get there and required first understanding how to prove the adjunction inequality using trisections. In this talk, I will discuss the basics of symplectic trisections, their relation to the adjunction inequality and the minimal genus problem, as well as some potential applications.

It is tradition at BIRS to take a group photo at every meeting, which gets posted on the workshop web page. So please turn on your cameras for a screenshot of the Zoom Gallery.

We will have two breakout rooms for discussions after the talks.

In joint work with T. Lidman and R. Lipshitz, we show that for K a nullhomologous knot in a 3-manifold Y and Sigma(Y,K) a double cover of Y branched along K, there exists a spectral sequence related the Heegaard Floer homology of Sigma(Y,K) and Y, and a corresponding rank inequality for HFhat. This extends recent work of T. Large and previous work of R. Lipshitz, and S. Sarkar, and I.

I will discuss the context and solution of the rectangular peg problem: for every smooth Jordan curve and rectangle in the Euclidean plane, one can place four points on the curve at the vertices of a rectangle similar to the one given. The solution involves symplectic geometry in a surprising way. ‘Joint work with Andrew Lobb.

A useful tool to study a 3-manifold is the space of representations of its fundamental group into a Lie group. Any 3-manifold can be decomposed as the union of two handlebodies. Thus representations of the 3-manifold group into a Lie group can be obtained by intersecting representation varieties of the two handlebodies. Casson utilized this observation to define his celebrated invariant. Later Taubes introduced an alternative approach to define Casson invariant using more geometric objects. By building on Taubes' work, Floer refined Casson invariant into a 3-manifold invariant which is known as instanton Floer homology. The Atiyah-Floer conjecture states that Casson's original approach can be also used to define a graded vector space and the resulting invariant of 3-manifolds is isomorphic to instanton Floer homology. In this talk, I will discuss a variation of the Atiyah-Floer conjecture, which states that framed Floer homology (defined by Kronheimer and Mrowka) is isomorphic to symplectic framed Floer homology (defined by Wehrheim and Woodward). I will also discuss how techniques from symplectic topology could be useful to study framed Floer homology. This talk is based on a joint work with Kenji Fukaya and Maksim Lipyanskyi.

We will have two breakout rooms for discussions after the talks.

I will describe how the knot Floer homology of a knot K can be represented by a decorated collection of immersed curves in the marked torus. The surgery formula for knot Floer homology translates nicely to this setting: the Heegaard Floer homology HF^- of p/q surgery on K is given by the Lagrangian Floer homology of these immersed curves with a line of slope p/q. For a simplified “UV = 0” version of knot Floer homology, the analogous statements follow from earlier work with Rasmussen and Watson by passing through the bordered Floer homology of the knot complement, but a more direct approach allows us to capture the stronger “minus” invariant by adding decorations to the curves. Often recasting algebraic structures in terms of geometric objects in this way leads to new insights and results; I will mention some applications of this immersed curves framework, including obstructions to cosmetic surgeries.

The Instanton Floer chain complex is generated by flat connections on a principal SU(2)-bundle over, and the differential counts solutions to the Yang-Mills equation (known as instantons). The Heegaard Floer chain complex is generated by the intersection points of curves in a Heegaard diagram for Y and its differential counts solutions to the Cauchy-Riemann equation (known as pseudoholomorphic Whitney discs). In the talk I will show that these invariants are the same when the 3-manifold is surgery on S^3 along a torus knot. This is joint work with Tye Lidman and Christopher Scaduto.

Given a decomposition of a knot K into two four-ended tangles T and T', the (holonomy perturbed) traceless-SU(2)-character-variety functor produces Lagrangians R(T) and R(T') in the pillowcase P. Hedden, Herald and Kirk used this to define Pillowcase homology, conjecturally the symplectic counter-part of the singular instanton homology I(K). Important in their construction is how R(T) and its restriction to P are affected by “adding an earring”, a process used by Kronheimer and Mrowka to avoid reducibles. The object that governs this process turns out to be an immersed Lagrangian correspondence from pillowcase to itself. We will describe this correspondence in detail, and study its action on Lagrangians. In the case of the (4,5) torus knot, we will see that a correction term from the bounding cochains must be added. We will indicate a particular figure eight bubble which recovers the desired bounding cochain, as predicted by Bottman and Wehrheim. This is ioint work with G. Cazassus, C. Herald and P. Kirk.

We will have two breakout rooms for discussions after the talks.