Geometric structures on 3 and 4 manifolds
The purpose of this meeting is to bring together a diverse
range of
experts and earlycareer researchers working in smooth, contact, and
symplectic topology in
lowdimensions.
Points of focus will include contact and symplectic structures, knot
concordance, Engel structures and the relation of these to unifying
notions such as
rigidity and flexibility, and Floer–theoretic frameworks of study.
The workshop will have a strong contingent of earlycareer
researchers,
and this will be reflected in the format of talks, which will include
many
short talks. There will also be discussion sessions scheduled
throughout the week.
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 Ngraphs, Roger Casals
In this talk I will explore the combinatorics of Ngraphs and their
relation to Legendrian surfaces. First, I will discuss the singularity
theory of 2dimensional wavefronts in 3space. Then, I will introduce
the notion of an Ngraph and present new examples and computations of
Floertheoretic invariants. This is joint work with Eric Zaslow.
10:10  10:40 Algebraic torsion in higherdimensional
contact manifolds,
Agustin Moreno
Using the notion of algebraic torsion due to LatschevWendl, in this
talk we sketch the construction of an infinite family of contact
manifolds with finite and nonzero 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 ChekanovEliashberg
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 CassonGordon invariants,
Ana Lecuona
Homology groups provide bounds on the minimal number of handles needed
in any handle decomposition of a manifold. We will use CassonGordon
invariants to get better bounds in the case of 4dimensional rational
homology balls with boundary a given rational homology 3sphere. 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 noncyclic quotient surface singularities. As byproduct, 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
3manifolds, Jonathan Simone
Using a generalization of a widelyused result of LiscaMatic 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 3manifolds with first Betti
number equal to 1.

Tuesday June 12
9:00  9:50 Pretzel links, mutation, and the sliceribbon
conjecture,
Paolo Aceto
We show that the mutant 2component 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 sliceribbon
conjecture for 4stranded 2component pretzel links. In order to
distinguish mutant links up to concordance we consider 3fold 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 3manifolds bounding rational
homology balls,
Kyungbae Park
We give a complete classification of the spherical 3manifolds that
bound smooth rational homology 4balls. Furthermore, we determine the
order of spherical 3manifolds in the rational homology cobordism group
of rational homology 3spheres. To this end, we use constraints for
3manifolds 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 4manifolds is that in a 4manifold it is not always possible to
represent every middle dimensional homology class with a smoothly
embedded sphere. This is true even among the simplest 4manifolds:
$X_0(K)$ obtained by attaching an $0$framed 2handle to the 4ball
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 3manifolds can be obtained from
the standard contact structure on the 3sphere 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 MorseBott computational methods for computing the contact
homology of 3dimenseional 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
4manifolds bounded by a rational homology 3sphere, Dong Heon Choe
We show that, if a rational homology 3sphere Y bounds a positive
definite smooth 4manifold, then there are finitely many negative
definite lattices, up to the stableequivalence, which can be realized
as the intersection form of a smooth 4manifold 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 3manifolds 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
4manifolds, 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 Quasipositive links and Heegaard Floer
homology, Alberto
Cavallo
We use our generalization of the OzsváthSzabó tauinvariant to show
that the maximal selflinking number of a quasipositive link L can be
computed from a quasipositive braid
presentation of L.
Furthermore, we prove that if we consider a (combinatorially
defined) subfamily of quasipositive links then the slice genus of
a link is also determined by the tauinvariant.
Finally, we briefly talk about a similar result for strongly
quasipositive 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 CheegerGromov rho invariants and Heegaard Floer
dinvariants 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 3manifold
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 cooriented 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 Σgbundles over circle whose monodromies are
taken from some collection of
pseudoAnosov diffeomorphisms. We show the existence of tight contact
structure on
every closed 3manifold obtained via rational rsurgery 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 3manifolds
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 2sphere. 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
4manifolds by studying involutions on 4manifolds. 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 4sphere which are embedded
homeomorphically, but are not smoothly ambiently isotopic. A further
potential application of interest is being able to study
antisymplectic involutions on 4manifolds, and the corresponding
geography problem for symplectic 4manifolds admitting such
involutions.
10:25  11:15 Weinstein 6Manifolds and Embedding
Questions, Jeremy Van
HornMorris
We describe a framework for embedding Weinstein 4manifolds into
Weinstein 6manifolds using Lefschetz fibrations. The process also
reframes the "spun embeddings" of Etnyre and Lekili used to describe
embeddings of contact 3manifolds into contact 5manifolds. We'll
discuss obstructions as well as examples including Weinstein
6manifolds which contain all Weinstein 4manifolds 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 fourdimensional Stein domain has a Morse function
whose
regular level sets are contact threemanifolds. This allows us to study
complex curves via their intersection with these contact level sets,
where we can apply threedimensional tools. Using this perspective, we
give a braidtheoretic characterization of the links in Steinfillable
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 HornMorris.)
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 BarNatan'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
4genus 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 HornMorris, Arkansas
Funding
We gratefully acknowledge support from the EPSRC and the National
Science Foundation