Geometric structures on 3 and 4 manifolds

Logistics

Participants are encouraged to make travel and lodging reservations as soon as is practical, since Dubrovnik is a popular tourist destination and hotel rooms can become scarce. Lodging options are fairly numerous, through travel sites such as booking.com or airbnb.com.

Schedule of Talks

• Monday June 11
  
  9:00 - 9:50 Flag moduli and N-graphs, Roger Casals
  In this talk I will explore the combinatorics of N-graphs and their relation to Legendrian surfaces. First, I will discuss the singularity theory of 2-dimensional wavefronts in 3-space. Then, I will introduce the notion of an N-graph and present new examples and computations of Floer-theoretic invariants. This is joint work with Eric Zaslow.
  
  10:10 - 10:40 Algebraic torsion in higher-dimensional contact manifolds, Agustin Moreno
  Using the notion of algebraic torsion due to Latschev-Wendl, in this talk we sketch the construction of an infinite family of contact manifolds with finite and non-zero order of torsion, in any odd dimension. It follows that they are tight and admit no strong symplectic fillings.
  
  11:10 - 12:00 Categories associated to Legendrian links, Baptiste Chantraine
  We will review the construction of two categories associated to legendrian link: the category of sheaves with micro support on the link (a dg categories) and the representation category (an A_infinity category) whose objects are representations of the Chekanov-Eliashberg dga of the link). We will then report on some joint work with L. Ng and S. Sivek where we compare the two categories for the rainbow closure of the (2,m) link.
  
  15:00 - 15:50 Complexity and Casson-Gordon invariants, Ana Lecuona
  Homology groups provide bounds on the minimal number of handles needed in any handle decomposition of a manifold. We will use Casson-Gordon invariants to get better bounds in the case of 4-dimensional rational homology balls with boundary a given rational homology 3-sphere. This analysis can be used to understand the complexity of the discs associated to ribbon knots in S^3. This is a joint work with P. Aceto and M. Golla.
  
  16:25 - 16:55 A Lefschetz fibration structure on minimal symplectic fillings of a quotient surface singularity, Hakho Choi
  In this talk we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration structure on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As by-product, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained by a sequence of rational blowdowns from its minimal resolution.
  
  17:10 - 18:00 Tight contact structures with no Giroux torsion on plumbed 3-manifolds, Jonathan Simone
  Using a generalization of a widely-used result of Lisca-Matic along with standard techniques from convex surface theory, we will classify the contact structures with no Giroux torsion (most of which are Stein fillable) on a certain class of plumbed 3-manifolds with first Betti number equal to 1.
  
  
  
• Tuesday June 12
  
  9:00 - 9:50 Pretzel links, mutation, and the slice-ribbon conjecture, Paolo Aceto
  We show that the mutant 2-component pretzel links P(p,q,-q,-p) and P(p,q,-p,-q) are not concordant for any distinct odd integers p and q greater than 1. As a corollary, we obtain a proof of the slice-ribbon conjecture for 4-stranded 2-component pretzel links. In order to distinguish mutant links up to concordance we consider 3-fold branched covers and use an obstruction based on Donaldson's diagonalization theorem. This is joint work with Min Hoon Kim, JungHwan Park and Arunima Ray.
  
  10:25 - 11:15 Spherical 3-manifolds bounding rational homology balls, Kyungbae Park
  We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology balls, induced from Donaldson's diagonalization theorem and Heegaard Floer correction terms. This is joint work with Dong Heon Choe.
  
  11:30 - 12:20 Shake genus and slice genus, Lisa Piccirillo
  An important difference between high dimensional smooth manifolds and smooth 4-manifolds is that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X_0(K)$, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves Problem 1.41 of the Kirby list. As corollaries we show that Rasmussen's $s$ invariant is not a $0$-trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus.
  
  
• Wednesday June 13
  
  9:00 - 9:50 Fillability of positive contact surgeries and Lagrangian disks, Bulent Tosun
  : It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically/Stein fillable when r is in (0,1]. This is joint work with James Conway and John Etnyre.
  
  10:25 - 11:15 Homotopy type of the space of embeddings of circles into $\R^4$ via Engel geometry, Fran Presas
  We reprove a classical result stating that the space of embeddings of circles into $R^4$ is simply connected. We do it by actually computing the fundamental group of the space of horizontal circles embeddings for the standard Engel structure (circles tangent to the Engel structure). We show how to generalize the proof to try to compute higher homotopy groups of that space.
  
  11:30 - 12:20 Reeb dynamics of the links of simple singularities, Jo Nelson
  We discuss Morse-Bott computational methods for computing the contact homology of 3-dimenseional prequantization bundles and applications to quantitative questions in dynamics. Of particular interest will be work in progress establishing a Floer theoretic McKay type correspondence for links of simple singularities via these geometric methods. Along the way, we will see a neat relationship between the Reeb dynamics and the presentation of these links as Seifert fiber spaces.
  
  
  15:00 - 15:50 Arboreal singularities and loose Legendrians, Emmy Murphy
  Suppose W is a Stein manifold which is diffeomorphic to C^n, and that the wrapped Fukaya category of W vanishes. Does it follow that W is symplectomorphic to C^n? Presently this question is well out of reach. We discuss some new partial results, in all dimensions but particularly n=3. The main ingredients are arboreal singularities -- a class of Lagrangian singularities which are simple enough to study but robust enough to apply in generality, microlocal sheaves -- an effective stand in for Fukaya categories which are better adapted for singular Lagrangians and combinatorics, and loose Legendrians -- a concept from contact flexibility which allows us to deduce geometric conclusions from topological hypotheses.
  
  16:25 - 16:55 On intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere, Dong Heon Choe
  We show that, if a rational homology 3-sphere Y bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by Y. To this end, we make use of constraints on definite forms bounded by Y induced from Donaldson's diagonalization theorem, and correction term invariants due to Ozsvath and Szabo. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.
  
  17:10 - 18:00 Symplectic isotopy problems, Laura Paul Starkston
  We will discuss some problems and results about symplectic surfaces in 4-manifolds, particularly in the complex projective plane. The main question is to classify symplectic surfaces up to symplectic isotopy. If the surface has singularities, we restrict the isotopies to the class of surfaces with the same model singularities.
  
  
  
• Thursday June 14
  
  9:00 - 9:50 Genus one Lefschetz fibrations on disk cotangent bundles of surfaces, Burak Ozbagci
  We describe a Lefschetz fibration of genus one on the disk cotangent bundle of any closed orientable surface S. As a corollary, we obtain an explicit genus one open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of S. Using a different method, we also describe a Lefschetz fibration of genus one on the disk tangent bundle of S and show that it is isomorphic to the Lefschetz fibration above.
  
  10:25 - 10:55 Quasi-positive links and Heegaard Floer homology, Alberto Cavallo
  We use our generalization of the Ozsváth-Szabó tau-invariant to show that the maximal self-linking number of a quasi-positive link L can be computed from a quasi-positive braid presentation of L. Furthermore, we prove that if we consider a (combinatorially defined) subfamily of quasi-positive links then the slice genus of a link is also determined by the tau-invariant. Finally, we briefly talk about a similar result for strongly quasi-positive links.
  
  11:10 - 12:00 The bipolar filtration of topologically slice knots, Min Hoon Kim
  The bipolar filtration of Cochran, Harvey and Horn initiated the study of deeper structures of the smooth concordance group of the topologically slice knots. In this talk, I would like to give an introduction to the bipolar filtration, and prove that every graded quotient of the bipolar filtration has infinite rank. The proof uses higher order amenable Cheeger-Gromov rho invariants and Heegaard Floer d-invariants of infinitely many cyclic branched covers simultaneously. This is joint work with Jae Choon Cha.
  
  15:00 - 15:50 Heegaard Floer, taut foliations, and regions of rational surgery slopes, Sarah Rasmussen
  Recent tools make it possible to partition the space of rational Dehn surgery slopes for a knot (or in some cases a link) in a 3-manifold into domains over which the Heegaard Floer homology of the surgered manifolds behaves continuously as a function of slope. I will describe some techniques for determining the walls of discontinuity separating these domains, along with progress towards interpreting some aspects of this structure in terms of the behaviour of co-oriented taut foliations and/or tight contact structures. This talk draws on a combination of independent work, previous joint work with Jake Rasmussen, and work in progress with Rachel Roberts.
  
  16:25 - 16:55 Tight contact structures on some hyperbolic three manifolds, Merve Secgin
  Let Σg denote a closed orientable surface of genus g ≥ 2. We consider a certain infinitefamily of Σg-bundles over circle whose monodromies are taken from some collection of pseudo-Anosov diffeomorphisms. We show the existence of tight contact structure on every closed 3-manifold obtained via rational r-surgery along a section of any member of the family whenever r 6= 2g − 1. Combining with Thurston’s hyperbolic Dehn surgery theorem, we obtain infinitely many hyperbolic closed 3-manifolds admitting tight contact structures.
  
  17:10 - 18:00 Rational cuspidal curves and symplectic fillings, Marco Golla
  A rational cuspidal curve is a complex curve whose singularities are cones over knots, and that are homeomorphic to a 2-sphere. These are more elusive objects than one could expect, and, for instance, their classification in the projective plane is not yet complete. I will discuss a symplectic perspective on the problem and some results obtained with Laura Starkston (in progress).
  
  
• Friday June 15
  
  9:00 - 9:50 Irreducible Embeddings and Equivariant Twisted Fiber Sums, Andrew Havens
  We discuss methods for constructing interesting surface embeddings in 4-manifolds by studying involutions on 4-manifolds. Drawing on the work of Finashin, Kreck and Viro in the 1980’s, as well as recent work of Nermin Salepci on Real Lefschetz fibrations, the initial goal is the construction of families of surfaces in the 4-sphere which are embedded homeomorphically, but are not smoothly ambiently isotopic. A further potential application of interest is being able to study anti-symplectic involutions on 4-manifolds, and the corresponding geography problem for symplectic 4-manifolds admitting such involutions.
  
  10:25 - 11:15 Weinstein 6-Manifolds and Embedding Questions, Jeremy Van Horn-Morris
  We describe a framework for embedding Weinstein 4-manifolds into Weinstein 6-manifolds using Lefschetz fibrations. The process also reframes the "spun embeddings" of Etnyre and Lekili used to describe embeddings of contact 3-manifolds into contact 5-manifolds. We'll discuss obstructions as well as examples including Weinstein 6-manifolds which contain all Weinstein 4-manifolds as properly embedded Weinstein submanifolds. This is joint work with Amey Kaloti.
  
  11:30 - 12:20 Complex curves through a contact lens, Kyle Hayden
   Every four-dimensional Stein domain has a Morse function whose regular level sets are contact three-manifolds. This allows us to study complex curves via their intersection with these contact level sets, where we can apply three-dimensional tools. Using this perspective, we give a braid-theoretic characterization of the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work in progress with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)
  
  15:00 - 15:50 Link homology and Floer homology in pictures, Adam Saltz
  There are no fewer than eight link homology theories which admit spectral sequences from Khovanov homology. These theories have very different origins -- representation theory, gauge theory, symplectic topology -- so it's natural to ask for some kind of unifying theory. I will attempt to describe this theory using Bar-Natan's pictorial formulation of link homology. This strengthens a result of Baldwin, Hedden, and Lobb and proves new functoriality results for several link homology theories. It may also be useful for studying mutation. (I won't assume much specific knowledge of these link homology theories!)
  
  16:25 - 16:55 Concordance bounds on the Thurston norm, Daniele Celoria
  We will show that the fully twisted correction terms in HF provide lower bounds on the Thurston norm of certain homology classes of a link complement, up to concordance. We then specialise this procedure to knots in S2xS1 , and obtain a lower bound on their geometric winding number. (joint work with M. Golla)
  
  17:10 - 18:00 Satellite operators on knot concordance, Allison Miller
  The set of knots modulo (smooth or topological) concordance can be considered with a variety of extra structures, such as the group structure induced by connected sum and the metric induced by the 4-genus function. Since the classical satellite construction behaves nicely with respect to concordance (if K and K' are concordant then P(K) and P(J) are concordant for any pattern P), it is natural to ask about the properties of satellite actions with respect to these extra structures. I will briefly survey known results in this area, and then discuss a recent result which gives a complete characterization of when two patterns induce operators which are a bounded distance from each other in terms of winding number.

Registration / Expense Reimbursement

Registration is now closed.

Expenses form (UK payments) Email to Andy.Wand@glasgow.ac.uk
Expenses form (Non US, non UK payments) Email to Andy.Wand@glasgow.ac.uk

US Participants: Contact the organisers at dubrovniktopology2018@gmail.com

Confirmed Participants

‎
• Paolo Aceto, MPIM
• Vitalijs Brejevs, Glasgow
• Roger Casals, MIT
• Alberto Cavallo, CEU
• Daniele Celoria, Oxford
• Baptiste Chantraine, Nantes
• Dong Heon Choe, Seoul National University
• Marco Golla, CNRS, Nantes
• Andrew Havens, University of Massachussets
• Kyle Hayden, Boston College
• Min Hoon Kim, KIAS
• Ana Lecuona, Marseille
• Irena Matkovič, CEU
• Allison Miller, Texas
• Agustin Moreno, Humboldt Universitat zu Berlin
• Emmy Murphy, Northwestern
• Jo Nelson, Rice
• Burak Ozbagci, Koc
• Hakho Choi, Seoul National University
• Kyungbae Park, KIAS
• Lisa Piccirillo, Texas
• Fran Presas, ICMAT
• Sarah Rasmussen, Cambridge
• Merve Secgin, METU
• Adam Saltz, Georgia
• Jon Simone, Virginia
• Laura Starkston, Stanford
• Bulent Tosun, Alabama
• Jeremy Van Horn-Morris, Arkansas

Organisers

Inanc Baykur, Tom Mark, Gordana Matić, Andy Wand

Funding

We gratefully acknowledge support from the EPSRC and the National Science Foundation