All talks will be in Skiles 006.
The conference will begin with one talk at 4:00 on Friday (preceded by a tea) and end by around 1:00 on Sunday.




3:304:00 

Tea 


4:004:50 

Colloquium: Dusa McDuff 

Staircases and cuspidal curves in symplectic four manifolds

5:006:00 

Lightning talk session 1 

9 talks see below
Slides

6:30 

Social Gathering 

Club Room at the Georgia Tech Hotel





9:009:30 

Bagels and light refreshments 


9:3010:20 

Talk: Kyle Hayden 

An atomic approach to Walltype stabilization problems Scanned Notes 
10:2011:00 

Tea Break 


11:0011:50 

Talk: Orsola CapovillaSearle 

Results on exact Lagrangian fillings of Legendrian links
Scanned Notes 
11:5012:30 

Catered Lunch 

Fox Bros BBQ 
12:301:30 

Lighting talk session 2 

9 talks see below
Slides

1:302:00 

break 


2:002:50 

Talk: Jo Nelson 

Positive torus knotted Reeb dynamics in the tight 3sphere
Scanned Notes 
2:504:10 

Extended Tea Break 


4:105:00 

Talk: Austin Christian 

Stably Weinstein Liouville domains
Slides 
6:00 

Banquet 

At: South City Kitchen





9:009:30 

Sublime Doughnuts and light refreshments 


9:3010:20 

Talk: Jake Rasmussen 

Parabolic Representations and Signatures Slides

10:2011:00 

Tea Break 

11:0011:50 

Talk: Patricia Cahn 

Algorithms for Computing Invariants of Trisected Branched Covers of the 4Sphere

 James Hughes  A NielsenThurston classification of Legendrian loops 
 Sierra Knavel  Fundamental groups of genus2 Lefschetz fibrations 
 Justin Murray  On the homotopy cardinality of the representation category 
 Shunyu Wan  Negative contact surgery and Legendrian nonsimple knots 
 Arka Banerjee  On the thickening of finite complexes into manifolds 
 Riccardo Pedrotti  Towards a count of holomorphic sections of Lefschetz fibrations over the disc 
 Alex Nolte  Foliations of Domains of Discontinuity for PSL(4,R) Hitchin Representations 
 Rob McConkey  Crossing Numbers of Cable Knots 
 Han Lou  On the Hofer Zehnder Conjecture for Semipositive Symplectic Manifolds 
 Katherine Raoux  What is rational 4genus? 
 Surena Hozoori  Anosov Reeb flows, Dirichlet optimization and entropy 
 Yikai Teng  Infinite order knot traces 
 Randy Van Why  Compactifying Symplectic 4Manifolds 
 Geunyoung Kim  Heegaard diagrams of 5manifolds 
 Daniel Hartman  Smoothing 3balls in the 5ball 
 Louisa Liles  Constructing Annular Links from Thompson's Group T 
 Roman Krutowski  Heegaard Floer symplectic homology and the generalized Viterbo's isomorphism theorem 
 Terrin Warren  Cork theorem for diffeomorphisms 
Titles and Abstracts
Algorithms for Computing Invariants of Trisected Branched Covers of the 4Sphere
Branched covers are a rich source of invariants of knotted surfaces in dimension 4. When a surface is in bridge position with respect to the genus0 trisection of the 4sphere, one can describe all branched covers of the sphere along that surface by permutationlabellings of a corresponding triplane diagram. Moreover, these branched covers are naturally trisected. We give diagrammatic algorithms for computing the group trisection, homology groups, and intersection form of a branched cover of S^4 along a bridgetrisected surface, given a permutationlabelled triplane diagram. We then apply our algorithm to dihedral and cyclic covers of spun knots, cyclic covers of Suciu's family of ribbon knots with the same knot group, and an infinite family of irregular covers of the Stevedore diskdouble. As an application, we give an algorithm for computing Kjuchukova's homotopyribbon obstruction for a pcolorable knot, for arbitrary odd p, given an extension of that coloring over a ribbon surface in the 4ball. This is joint work with Gordana Matic and Ben Ruppik.
Stably Weinstein Liouville domains
Morse functions are generic among realvalued functions on a smooth manifold, and distinct Morse functions on a fixed manifold are related to each other in wellunderstood ways. In the contact and symplectic categories, Morse theory is more delicate and certainly less wellunderstood. In this talk we will address the question of whether an exact symplectic manifoldwithboundary can be made compatible with a Morse function  that is, whether a Liouville domain can be made Weinstein. We provide some explicit constructions for realizing this compatibility, and use these to demonstrate that a famous family of LiouvillebutnotWeinstein domains first studied by Geiges and Mitsumatsu is stably Weinstein. This talk is based on joint work with Joseph Breen.
Results on exact Lagrangian fillings of Legendrian links
An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about new connections between fillings and Newton polytopes, as well as results on distinguishing nonorientable fillings. This is based on joint work with Casals and joint work with Hughes and Weng.
An atomic approach to Walltype stabilization problems
In dimension four, the differences between continuous and differential topology are vast but fundamentally unstable, disappearing when manifolds are enlarged in various ways. Walltype stabilization problems aim to quantify this instability. I will discuss an approach to these problems that considers the basic building blocks of hcobordisms and uses this to construct exotic 4manifolds, knotted surfaces, and other objects that are candidates to solve Walltype problems. In particular, as a proof of concept, I will exhibit exotically knotted surfaces in the 4ball that survive "internal" stabilization, as detected using Khovanov homology.
Staircases and cuspidal curves in symplectic four manifolds
This talk will give an elementary introduction to my joint work with Kyler Siegel that shows how cuspidal curves in a symplectic manifold X such as the complex projective plane determine when an ellipsoid can be symplectically embedded into X.
Positive torus knotted Reeb dynamics in the tight 3sphere
I will discuss joint work with Morgan Weiler on the Calabi invariant of periodic orbits of symplectomorphisms of Seifert surfaces of torus knots in the standard contact 3sphere. Our results come by way of spectral invariants of embedded contact homology, from which we establish quantitative action and linking existence results for periodic Reeb orbits with respect to a positive torus knotted elliptic orbit in the standard contact 3sphere. Along the way, we developed new methods for understanding embedded contact homology of open books and seifert fiber spaces.
Parabolic Representations and Signatures
An old conjecture of Riley, subsequently proved by Gordon, states that the fundamental group of a rational knot K has at least sigma(K)/2 real parabolic representations where sigma denotes the knot signature. I'll discuss an invariant of small knots in S^3 which can be interpreted as a signed count of real parabolic representations. As a corollary, we show that a version of Riley's conjecture holds for many other small knots, including small alternating and Montesinos knots. This is joint work with Nathan Dunfield.
Lightning talk
On the thickening of finite complexes into manifolds Given an aspherical simplicial complex, what is the minimum dimension of a manifold that is homotopy equivalent to that complex? Does this number remain the same if we take finite index cover of the complex? I will describe a new class of finite aspherical complexes where the answer to the second question is no.
Smoothing 3balls in the 5ball Derived from the work of Freedman and Quinn, it is known that any smooth 2knot in the 4sphere with complement having infinite cyclic fundamental group bounds a locally flat 3ball. However, a significant open question persists: do the 2spheres bound a smooth 3ball? An easier question to ask is whether the 2sphere is smoothly slice, an assertion proven by Kervaire. Consequently, every 3ball in the 4sphere is homotopic to a smooth 3ball in the 5ball relative to the 2knot. Between these extremes lies the question of whether the topological 3ball can be isotoped to a smooth 3ball in the 5ball. It turns out that the answer to this question depends entirely on the Rochlin invariant of the 2knot.
Anosov Reeb flows, Dirichlet optimization and entropy Given an almost complex structure on a contact 3manifold, Chern and Hamilton ask when the minimum of the associated Dirichlet energy functional is achieved. We classify all the minimizers of such energy functional, giving a complete answer to ChernHamilton question in dimension 3. Furthermore, in the case of Anosov Reeb flows, we show that the infimum of Dirichlet energy functional is closely related to Reeb dynamics and can be computed in terms of its entropy.
A NielsenThurston classification of Legendrian loops Legendrian loops are Legendrian isotopies of Legendrian links that fix the link setwise at time one. The trace of such a Legendrian isotopy forms an exact Lagrangian concordance, acting on the set of exact Lagrangian surfaces with a specified Legendrian link at the boundary. In this talk, I will introduce a classification of Legendrian loops analogous to the NielsenThurston classification of mapping classes. I will then briefly describe an application of this classification to understanding fixed point properties of the induced action of Legendrian loops on the sheaf moduli of a Legendrian.
Heegaard diagrams of 5manifolds We introduce a version of Heegaard diagrams for 5manifolds, which we use to study closed 5manifolds as well as 4manifolds and cobordisms between them. As an application, we study Heegaard diagrams of 5dimensional cobordisms from the standard 4sphere to Gluck twists along knotted 2spheres. This gives some equivalent statements to Gluck twists being diffeomorphic to the standard 4sphere.
Fundamental groups of genus2 Lefschetz fibrations Lefschetz fibrations are fundamental objects in the study of symplectic 4manifolds. Genus2 Leschetz fibrations over the 2sphere are especially interesting because of the hyperelliptic structure they carry. We will discuss some preliminary results on what finitely presented groups can be realized as the fundamental group of a genus2 Lefschetz fibration over the 2sphere.
Heegaard Floer symplectic homology and the generalized Viterbo's isomorphism theorem We introduce a novel invariant of a Liouville domain M called Heegaard Floer symplectic cohomology HFSH_k^*(M). It arises by applying Liphitz's cylindrical reformulation heuristic of Heegaard Floer homology in the setting of an arbitrary Liouville domain. In the particular case of a cotangent bundle M=T^*Q, we show that there is an isomorphism between HFSH_k^*(T^*Q) and homology of a certain Morsetheoretic model of the free loop space of the kthe symmetric power Sym^k(Q), which generalizes Viterbo's isomorphism theorem to the problem of motion of k identical particles.
Constructing Annular Links from Thompson's Group T In 2014 Jones introduced a method of constructing links in the 3sphere from elements of Thompson's group F. I provide an analog of this program for annular links and Thompson's group T. The main result is that every edgesigned graph embedded in the annulus is the Tait graph of an annular link created from T.
On the Hofer Zehnder Conjecture for Semipositive Symplectic Manifolds Arnold conjecture says that the number of 1periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality doesn't hold. In this talk, I will show that the HoferZehnder conjecture is true for semipositive symplectic manifolds with semisimple quantum homology. This is a joint work with Marcelo Atallah.
Crossing Numbers of Cable Knots Given a satellite knot K and its companion knot C, it remains open whether the crossing number of K is greater than or equal to the crossing number of C. We look at a case of satellite knots for which we can give a direct computation for the crossing number with respect to its companion knot. We will also mention briefly how the colored Jones polynomial is used to make this calculation.
On the homotopy cardinality of the representation category Given a Legendrian knot in $(\R^3, \ker(dzydx))$ one can assign a combinatorial invariants called ruling polynomials. These invariants have been shown to recover not only a (normalized) count of augmentations but are also closely related to a categorical count of augmentations in the form of the homotopy cardinality of the augmentation category. In this talk, we will indicate that that the homotopy cardinality of the $n$dimensional representation category is a multiple of the $n$colored ruling polynomial. Along the way, we establish that two $n$dimensional representations are equivalent in the representation category if they are ``conjugate homotopic'' and provide some applications to Lagrangian concordance.
Foliations of Domains of Discontinuity for PSL(4,R) Hitchin Representations In a breakthrough 2008 paper, GuichardWienhard proved the PSL(4,R) Hitchin component of a closed surface group π_1(S) arises as holonomies of properly convex foliated projective structures on the unit tangent bundle of S. The foliations here come from groupinvariant foliations of a domain of discontinuity Ω in RP^3 by properly convex subsets of copies of RP^2 and RP^1. I'll talk about current work that shows there's exactly one other geometrically similar foliation in codimension 2, and the codimension1 foliation is unique. This leads to a classification of similar refinements to GuichardWienhard's of the projective structures induced by PSL(4,R) Hitchin representations by their action on RP^3. The proof relies on working out a detailed qualitative picture for the shape of Ω.
Towards a count of holomorphic sections of Lefschetz fibrations over the disc Given a positive factorisation of the identity in the mapping class group of a surface S, we can associate to it a Lefschetz fibration over the sphere with S as a regular fiber. Its total space X is a symplectic 4manifold, so it is a natural question to ask what kind of invariants of X can be read off from this construction: the word in Mod(S) leads to an easy computations of the homology of X and I. Smith pushed this further by providing us a formula for the signature of the total space in terms of this combinatorial construction. Using this as a motivation, I will report on an ongoing joint work with Tim Perutz, aimed at obtaining an explicit formula for counting holomorphic sections of a Lefschetz fibration over the disk, while keeping track of their relative homology classes. By taking the monodromy of the fibration to be isotopic to the identity, we should get a count of sections for a Lefschetz fibration over the sphere, and in particular an invariant of its total space X. Thanks to Taubes’ SW=GW, these invariants should be closely related to the Seiberg Witten invariants of X.
What is rational 4genus? I will present a definition and explain it. This is joint work with Matthew Hedden.
Infinite order knot traces In 2017, Robert Gompf proved the existence of infinite order corks~\cite{Gom17}. Namely, there exists a cork $(C,f)$ such that $f^k$ does not extend to a selfdiffeomorphism of the cork for any $k$. In fact, if we do a cork twist via such maps $f^k$, it is possible to result in pairwise nondiffeomorphic 4manifolds. In this talk, we will adopt this idea to show that knot traces can actually behave as infinite order plugs.
Compactifying Symplectic 4Manifolds I will discuss compactifications of open symplectic 4manifolds. I will discuss some conditions for the existence of certain “nice” compactifications and perhaps some statements about the collection of all such spaces.
Negative contact surgery and Legendrian nonsimple knots We give infinitely many examples of Legendrian knots that have different Legendrian representatives with the same ThurstonBennequin number and rotation number such that any negative rational contact surgery along those representatives gives non contact isotopic 3manifolds.
Cork theorem for diffeomorphisms In dimension 4, there exist pairs of simply connected manifolds which are hcobordant but not diffeomorphic; these pairs of manifolds are related by cork twists. Similarly, there exist pairs of diffeomorphisms of simple connected 4 manifolds which are pseudoisotopic but not smoothly isotopic. In this talk, I will talk about how these two questions are related and discuss some preliminary results towards a “cork theorem” for diffeomorphisms.
