Schedule & Abstracts

9:00 - 9:50  David Gabai 

10:10 - 11:00  Daniel Hartman 

Coffee

11:30 - 12:20  Daniel Galvin

Lunch

2:30 - 3:20  Juan Muñoz-Echániz

Coffee

4:00 - 4:50  Boyu Zhang

6  Conference Dinner

9:00 - 9:50  Agniva Roy

10:10 - 11:00  Melissa Zhang

Coffee

11:30 - 12:20  Qiuyu Ren

Lunch

2:30 - 3:20  Alexandra Kjuchukova

Coffee

4:00 - 4:50  Alex Zupan

David Gabai

Loops of spheres in \#_2 S^2\times S^2

We will discuss loops of two 2-spheres in \#_2 S^2\times S^2, where the i'th 2-sphere is based at the i'th pt\times S^2. Joint work with David Gay and Daniel Hartman. 

Daniel Galvin

External stable smooth isotopy of surfaces in 4-manifolds

Many distinctions between smooth and topological phenomena disappear after we allow stabilisations, i.e. taking connected-sums with copies of $S^2 \times S^2$.  Wall proved that homotopy equivalent, simply-connected, smooth 4-manifolds are stably diffeomorphic, and this was extended by Gompf and Kreck to show that homeomorphic, orientable, smooth 4-manifolds are stably diffeomorphic.  We consider a related notion for embeddings of surfaces: two smoothly embedded surfaces are external-stably smoothly isotopic if they become smoothly isotopic after stabilising the ambient manifold some number of times.  We show that topological isotopy implies external stable smooth isotopy in many cases.  This is joint work with Patrick Orson and Mark Powell. 

Daniel Hartman

A combinatorial approach to families of surfaces in 4-manifolds

In Quinn’s foundational paper on pseudo-isotopes, he pioneered the notion that the complexity of a 1-parameter family of spheres relative to a fixed embedded surface could be understood from some combinatorial data; a finger/Whitney system. The finger/Whitney system gives a combinatorial representation of a path of a family of surfaces in a 4-manifold. In this talk, I will discuss the ambiguity associated with these combinatorial representatives. 

Alexandra Kjuchukova

Framed knot concordance and signatures

We define a new integer-valued invariant of framed concordance between certain framed knots in closed oriented 3-manifolds. As an application, we derive an equation relating the signatures of various 4-manifolds which arise as branched covers of surfaces in $B^4$, the groups of whose exteriors admit dihedral quotients. The signature formula admits an interpretation in which the terms are various classical knot invariants. The talk is based on joint work with Julius Shaneson.

Juan Muñoz-Echániz

Constraints on Lefschetz fibrations with 4-dimensional fibers

I will describe a new constraint on the topology of Lefschetz fibrations with 4-dimensional fibers, arising from the analysis of the framed bordism class of Seiberg–Witten moduli spaces. This constraint yields smooth isotopy obstructions for compositions of Dehn twists along self-intersection -2 spheres in 4-manifolds. As an application, we give a negative answer to a question of Donaldson asking whether, for a closed simply-connected symplectic 4-manifold, the symplectic Torelli group is generated by squared Dehn twists on Lagrangian spheres. Based on joint work with H. Konno, J. Lin, and A. Mukherjee.

Qiuyu Ren

Khovanov skein lasagna modules with 1-dimensional inputs

Skein lasagna modules (with 0-dimensional inputs) for smooth 4-manifolds were introduced by Morrison--Walker--Wedrich, and have found applications in detecting exotic phenomena in dimension 4. In this talk, we introduce skein lasagna modules with 1-dimensional inputs, which takes in a suitably functorial homology theory for links in connect sums of S^1\times S^2, and produces smooth 4-manifold invariants. The Khovanov homology in #S^1\times S^2 introduced by Rozansky and Willis is shown to be sufficiently functorial to supply such an input theory. We touch upon some ingredients to the proof, including an isomorphism of Sullivan--Zhang relating the Khovanov skein lasagna modules (with 0-dimensional inputs) to Rozansky--Willis's invariant, a calculation of the diffeomorphism group modulo local diffeomorphism for a class of 4-manifolds, and a Gluck twist construction in Khovanov skein lasagna modules (with 0-dimensional inputs).

Agniva Roy

Spinal open books and fillings of some L-spaces

Understanding symplectic fillings of contact 3-manifolds is a problem with a rich history. It has connections to singularity theory and smoothings of complex singularities, smooth low-dimensional topology and exotic structures, mapping class groups of surfaces, as well as studying the interfaces between smooth, symplectic, Liouville, and Stein structures on manifolds. In recent joint work with Hyunki Min and Luya Wang, we extend a program of Wendl and Lisi--Van Horn-Morris--Wendl to use holomorphic foliations to reduce the symplectic fillability and filling classification problem of ``partially planar'' 3-manifolds to a monodromy factorisation problem. In current work, we apply these techniques to classify fillings of L-spaces obtained by surgery on certain torus knots. In this talk, we will see an overview of how such holomorphic foliation techniques have been used in symplectic topology, and see applications to new classification results.

Boyu Zhang

Dax invariants, light bulbs, and isotopies of symplectic structures 

In this talk, I will present several results about isotopy problems in dimension 4.  First, we give a classification of the isotopy classes of embeddings of $\Sigma$ in $\Sigma\times S^2$ that are geometrically dual to $\{pt\}\times S^2$, where $\Sigma$ is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic. This answers a question of Gabai. Second, we show that the space of symplectic forms on an irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components. This gives the first such example among closed 4-manifolds and answers a question of McDuff-Salamon.  We also show that symplectic forms on a closed 4-manifold with a fixed cohomology class do not admit the h-principle, which answers a question of Cieliebak-Eliashberg-Mishachev.  The proofs are based on a generalization of the Dax invariant to embedded closed surfaces.  This is joint work with Jianfeng Lin, Weiwei Wu, and Yi Xie.

Melissa Zhang

Skein lasagna modules and Rozansky-Willis homology

In this talk I will describe joint work with Ian Sullivan, where we use properties of categorified projectors to prove that the Khovanov skein lasagna module of $S^2 \times S^2$ is trivial. Along the way, we will discover a relationship between the skein lasagna module of $S^2 \times D^2$ with a link $L$ in the boundary and the Rozansky-Willis homology of $L$ inside $S^2 \times S^1$. This isomorphism is used in recent joint work with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Michael Willis, where we define a new version of $gl_2$ skein lasagna modules with 1-dimensional inputs.

Alex Zupan

A homological generalized property R conjecture is false

Gabai proved that knots have property R, meaning that the only way to obtain S^1 x S^2 by Dehn surgery on a knot in the 3-sphere is to do 0-surgery on the unknot.  The generalized property R conjecture (GPRC) posits a similar situation for links, claiming that if L is an n-component link in the 3-sphere with a surgery to the connected sum of n copies of S^1 x S^2, then L must be handleslide equivalent to the unlink.  While there are many potential counterexamples to the GPRC in the literature, obstructing handleslide equivalence is a tricky proposition.  Thus, we further generalize the GPRC to include a mixture of geometric and homological data, and then we provide a counterexample to this broader version of the GPRC.  This is joint work with Tye Lidman and Trevor Oliveira-Smith.