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.

Lightning talks session 1:

Lightning talks session 2:

Titles and Abstracts

Patricia Cahn
Title: Algorithms for Computing Invariants of Trisected Branched Covers of the 4-Sphere
Abstract: 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 genus-0 trisection of the 4-sphere, one can describe all branched covers of the sphere along that surface by permutation-labellings of a corresponding tri-plane 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 bridge-trisected surface, given a permutation-labelled tri-plane 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 disk-double. As an application, we give an algorithm for computing Kjuchukova's homotopy-ribbon obstruction for a p-colorable knot, for arbitrary odd p, given an extension of that coloring over a ribbon surface in the 4-ball. This is joint work with Gordana Matic and Ben Ruppik.

Austin Christian
Title: Stably Weinstein Liouville domains
Abstract: Morse functions are generic among real-valued functions on a smooth manifold, and distinct Morse functions on a fixed manifold are related to each other in well-understood ways. In the contact and symplectic categories, Morse theory is more delicate and certainly less well-understood. In this talk we will address the question of whether an exact symplectic manifold-with-boundary 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 Liouville-but-not-Weinstein domains first studied by Geiges and Mitsumatsu is stably Weinstein. This talk is based on joint work with Joseph Breen.

Orsola Capovilla-SearleName
Title: Results on exact Lagrangian fillings of Legendrian links
Abstract: 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 4-ball. 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 non-orientable fillings. This is based on joint work with Casals and joint work with Hughes and Weng.

Kyle Hayden
Title: An atomic approach to Wall-type stabilization problems
Abstract: In dimension four, the differences between continuous and differential topology are vast but fundamentally unstable, disappearing when manifolds are enlarged in various ways. Wall-type stabilization problems aim to quantify this instability. I will discuss an approach to these problems that considers the basic building blocks of h-cobordisms and uses this to construct exotic 4-manifolds, knotted surfaces, and other objects that are candidates to solve Wall-type problems. In particular, as a proof of concept, I will exhibit exotically knotted surfaces in the 4-ball that survive "internal" stabilization, as detected using Khovanov homology.

Dusa McDuff
Title: Staircases and cuspidal curves in symplectic four manifolds
Abstract: 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.

Jo Nelson
Title: Positive torus knotted Reeb dynamics in the tight 3-sphere
Abstract: 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 3-sphere. 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 3-sphere. Along the way, we developed new methods for understanding embedded contact homology of open books and seifert fiber spaces.

Jake Rasmussen
Title: Parabolic Representations and Signatures
Abstract: 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

Arka Banerjee
Title: On the thickening of finite complexes into manifolds
Abstract: 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.

Daniel Hartman
Title: Smoothing 3-balls in the 5-ball
Abstract: Derived from the work of Freedman and Quinn, it is known that any smooth 2-knot in the 4-sphere with complement having infinite cyclic fundamental group bounds a locally flat 3-ball. However, a significant open question persists: do the 2-spheres bound a smooth 3-ball? An easier question to ask is whether the 2-sphere is smoothly slice, an assertion proven by Kervaire. Consequently, every 3-ball in the 4-sphere is homotopic to a smooth 3-ball in the 5-ball relative to the 2-knot. Between these extremes lies the question of whether the topological 3-ball can be isotoped to a smooth 3-ball in the 5-ball. It turns out that the answer to this question depends entirely on the Rochlin invariant of the 2-knot.

Surena Hozoori
Title: Anosov Reeb flows, Dirichlet optimization and entropy
Abstract: Given an almost complex structure on a contact 3-manifold, 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 Chern-Hamilton 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.

James Hughes
Title: A Nielsen-Thurston classification of Legendrian loops
Abstract: 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 Nielsen-Thurston 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.

Geunyoung Kim
Title: Heegaard diagrams of 5-manifolds
Abstract: We introduce a version of Heegaard diagrams for 5-manifolds, which we use to study closed 5-manifolds as well as 4-manifolds and cobordisms between them. As an application, we study Heegaard diagrams of 5-dimensional cobordisms from the standard 4-sphere to Gluck twists along knotted 2-spheres. This gives some equivalent statements to Gluck twists being diffeomorphic to the standard 4-sphere.

Sierra Knavel
Title: Fundamental groups of genus-2 Lefschetz fibrations
Abstract: Lefschetz fibrations are fundamental objects in the study of symplectic 4-manifolds. Genus-2 Leschetz fibrations over the 2-sphere 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 genus-2 Lefschetz fibration over the 2-sphere.

Roman Krutowski
Title: Heegaard Floer symplectic homology and the generalized Viterbo's isomorphism theorem
Abstract: 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 Morse-theoretic model of the free loop space of the k-the symmetric power Sym^k(Q), which generalizes Viterbo's isomorphism theorem to the problem of motion of k identical particles.

Louisa Liles
Title: Constructing Annular Links from Thompson's Group T
Abstract: In 2014 Jones introduced a method of constructing links in the 3-sphere 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 edge-signed graph embedded in the annulus is the Tait graph of an annular link created from T.

Han Lou
Title: On the Hofer Zehnder Conjecture for Semipositive Symplectic Manifolds
Abstract: Arnold conjecture says that the number of 1-periodic 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 Hofer-Zehnder conjecture is true for semipositive symplectic manifolds with semisimple quantum homology. This is a joint work with Marcelo Atallah.

Rob McConkey
Title: Crossing Numbers of Cable Knots
Abstract: 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.

Justin Murray
Title: On the homotopy cardinality of the representation category
Abstract: Given a Legendrian knot in $(\R^3, \ker(dz-ydx))$ 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.

Alex Nolte
Title: Foliations of Domains of Discontinuity for PSL(4,R) Hitchin Representations
Abstract: In a breakthrough 2008 paper, Guichard-Wienhard 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 group-invariant 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 codimension-1 foliation is unique. This leads to a classification of similar refinements to Guichard-Wienhard'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.

Riccardo Pedrotti
Title: Towards a count of holomorphic sections of Lefschetz fibrations over the disc
Abstract: 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 4-manifold, 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.

Katherine Raoux
Title: What is rational 4-genus?
Abstract: I will present a definition and explain it. This is joint work with Matthew Hedden.

Yikai Teng
Title: Infinite order knot traces
Abstract: 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 self-diffeomorphism 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 non-diffeomorphic 4-manifolds. In this talk, we will adopt this idea to show that knot traces can actually behave as infinite order plugs.

Randy Van Why
Title: Compactifying Symplectic 4-Manifolds
Abstract: I will discuss compactifications of open symplectic 4-manifolds. I will discuss some conditions for the existence of certain “nice” compactifications and perhaps some statements about the collection of all such spaces.

Shunyu Wan
Title: Negative contact surgery and Legendrian non-simple knots
Abstract: We give infinitely many examples of Legendrian knots that have different Legendrian representatives with the same Thurston-Bennequin number and rotation number such that any negative rational contact surgery along those representatives gives non contact isotopic 3-manifolds.

Terrin Warren
Title: Cork theorem for diffeomorphisms
Abstract: In dimension 4, there exist pairs of simply connected manifolds which are h-cobordant 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.