Georgia Institute of Technology
December 7 to 9, 2018

Schedule

 Friday 3:30-4:00 Tea 4:00-4:50 Talk:Tara Brendle* Congruence subgroups of braid groups 5:05-5:50 Lightning talk session 1 7 talks see below Slides 6:00- Social Gathering Publik Draft House Saturday 9:00-9:30 Light refreshments 9:30-10:20 Talk: Andrew Putman The stable cohomology of the moduli space of curves with level structures 10:20-11:00 Tea Break 11:00-11:50 Talk: Kevin Kordek The rational cohomology of the level 4 braid group 11:50-12:30 Catered Lunch Fox Bros BBQ 12:30-1:30 Lighting talk session 2 8 talks see below Slides 1:30-2:00 break 2:00-2:50 Talk: Sucharit Sarkar Khovanov homotopy typesScanned Notes 2:50-4:10 Extended Tea Break 4:10-5:00 Talk: Allison Miller Satellite operators on concordance Slides 6:00- Banquet At: South City Kitchen Sunday 9:00-9:30 Light refreshments 9:30-10:20 Talk: Juanita Pinzon Caicedo Satellites of Infinite Rank in ConcordanceScanned Notes 10:25-11:25 Lighting talk session 3 8 talks see below Slides 11:25-11:40 Tea Break 11:40-12:30 Talk: Gordana Matic Spectral order contact invariant Scanned Notes

All talks are in Skiles 006.

* This is a colloquium talk

Lightning talks session 1:

 JungHwan Park Concordance of knots in 3-manifolds Marissa Kawehi Loving Length spectra of q-differential metrics Linh Truong An infinite rank summand of the homology cobordism group Stephen McKean Enriching Bézout’s Theorem Moses Koppendrayer Quasipositive Surfaces and Convex Surface Theory Jacob Russell The Geometry of the Separating Curve Graph Christopher Anderson Splitting surfaces of 2-component links with multivariable Alexander polynomial 0

Lightning talks session 2:

 Justin Lanier Adding points to configurations in closed balls Sinem Onaran Cryptographic applications of braids Hyunki Min Contact structures on hyperbolic manifolds Adam Saltz Link homology, bridge trisections and knotted surface invariants Sangjin Lee Towards a Higher-dimensional construction of stable/unstable Lagrangian laminations Michael Dougherty Orthoscheme Configuration Spaces Maggie Miller Fibering ribbon disk complements Surena Hozoori Curvature, Contact Topology and Reeb Dynamics

Lightning talks session 3:

 Steve Trettel Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions Melissa Zhang Symmetry and Localization Haofei Fan Embedded Grid Homology William Olsen Trisecting Ozsvath-Szabo four-manifold invariants Mariano Echeverria Naturality of the Contact Invariant in Monopole Floer Homology under Strong Symplectic Cobordisms Sahana H Balasubramanya Acylindrical group actions on quasi-trees Joshua Pankau Constructing surfaces from positive integer matrices Yvon Verberne Shadows from the pure mapping class group to the curve graph

Titles and Abstracts

Tara Brendle
Title: Congruence subgroups of braid groups
Abstract: The Burau representation plays a key role in the classical theory of braid groups. When we let the complex parameter t take the value -1, we obtain a symplectic representation of the braid group known as the integral Burau representation. In this talk we will give a survey of results on braid congruence subgroups, that is, the preimages under the integral Burau representation of principal congruence subgroups of symplectic groups. Along the way, we will see the (perhaps surprising) appearance of braid congruence subgroups in a variety of other contexts, including knot theory, homotopy theory, number theory, and algebraic geometry.

Kevin Kordek
Title: Two theorems about braid groups
Abstract: In this first part of the talk, I will discuss joint work with Dan Margalit on the rational cohomology of the level 4 braid group, which is the kernel of the mod 4 reduction of the integral Burau representation. The main result is a formula for the first Betti number. In the second part of the talk, I will discuss joint work with Lei Chen and Dan Margalit on the classification of homomorphisms between braid groups. Our classification yields an explicit description of all possible homomorphisms from B_n to B_m with n at least 12 and m between n and 2n.

Gordana Matic
Title: Spectral order contact invariant
Abstract: We provide a refinement of the Ozsv\'ath - Szab\'o contact invariant by introducing a filtration into the complex that calculates it. The spectral order invariant takes values in $Z_{\geq 0} \cup \infty$, is zero for overtwisted contact structures, $\infty$ for Steinfillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. It gives a criterion for tightness of a contact structure stronger than the one given by the Heegaard Floer contact invariant, and provides an obstruction to existence of Stein cobordisms between contact 3-manifolds. We exhibit an infinite family of examples with vanishing Heegaard Floer contact invariant on which our invariant assumes an unbounded sequence of finite and non-zero values. This is joint work with \c Ca\u gatay Kutluhan, Jeremy Van Horn-Morris and Andy Wand.

Allison Miller
Title: Satellite operators on concordance
Abstract: The collection $\mathcal{C}$ of knots in $S^3$ modulo concordance has a variety of interesting structures. For example, connected sum of knots gives $\mathcal{C}$ the structure of an abelian group; there is a simple metric on $\mathcal{C}$ induced by the 4-genus; and there are several infinite filtrations of $\mathcal{C}$, most famously that of Cochran-Orr-Teichner. Every pattern, or knot in a solid torus, induces a well-defined satellite map on $\mathcal{C}$. Even basic questions such as when such a map is injective or surjective are generally open, but one can also ask how satellite-induced maps interact with these additional structures. I will survey the known results and open questions in this area, paying particular attention to the difference between the smooth and topological categories, and end by discussing in more detail the problem of when satellite maps induce group homomorphisms.

Juanita Pinzon Caicedo
Title: Satellites of Infinite Rank in Concordance
Abstract: Oriented knots are said to be concordant if they cobound an embedded cylinder in the interval times the 3-sphere. This defines an equivalence relation under which the set of knots becomes an abelian group with the connected sum operation. The importance of this group lies in its strong connection with the study of 4-manifolds. Indeed, many questions pertaining to 4-manifolds with small topology (like the 4-sphere) can be addressed in terms of concordance. A powerful tool for studying the algebraic structure of this group comes from satellite operations or the process of tying a given knot P along another knot K to produce a third knot P(K). In the talk I will describe how to use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group.

Andrew Putman
Title: The stable cohomology of the moduli space of curves with level structures
Abstract: I will prove that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as that of the ordinary moduli space of curves: a polynomial ring in the Miller-Morita-Mumford classes.

Sucharit Sarkar
Title: Khovanov homotopy types
Abstract: Various versions of Khovanov chain complexes are built from functors from the cube to Abelian groups. By lifting these functors to the Burnside category, one can construct CW complexes whose cellular chain complexes agree with the Khovanov complexes. I will present a general outline of this construction, focusing specifically on the even and odd Khovanov homology. The even theory is joint with Tyler Lawson and Robert Lipshitz, and the odd theory is joint with Chris Scaduto and Matt Stoffregen.

Lightning talk

Christopher Anderson
Title: Splitting surfaces of 2-component links with multivariable Alexander polynomial 0
Abstract: A 2-component link has multivariable Alexander polynomial 0 if and only if the second homology of the universal abelian cover of the link exterior is free on one generator when regarded as a module over the ring of integer coefficient Laurent polynomials in two variables. This generator may be represented by an essential surface in the universal abelian cover. When the generator may be represented by a sphere, the link must be a split link. When the generator may be represented by a torus but not a sphere, we show that the link is a toroidal boundary link.

Sahana H Balasubramanya
Title: Acylindrical group actions on quasi-trees
Abstract: A group G is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space - this is a very broad class of groups and contains many interesting examples. I show that every acylindrically hyperbolic group actually admits a non-elementary acylindrical action on a quasi-tree . A quasi-tree is a connected graph quasi-isometric to a tree, and forms a very special subclass of hyperbolic spaces. I will talk about the motivation behind this result and the tools used in the proof.

Michael Dougherty
Title: Orthoscheme Configuration Spaces
Abstract: Configuration spaces of manifolds provide natural examples of high-dimensional spaces with low-dimensional intuition. A combinatorial version of these spaces was presented by Abrams in 2000; the configuration space of a graph is defined to be a metric cube complex which Abrams shows is non-positively curved. In recent work with Jon McCammond and Stefan Witzel, we introduce a similar type of configuration space for directed graphs and, more generally, Delta-complexes. In this talk, I will define orthoscheme configuration spaces and present what is known so far.

Mariano Echeverria
Title: Naturality of the Contact Invariant in Monopole Floer Homology under Strong Symplectic Cobordisms
Abstract: The contact invariant is an element in the monopole Floer homology groups of a three manifold associated to a contact structure. A non-vanishing contact invariant implies tightness, so understanding its behavior under symplectic cobordisms is of interest to further exploit this property. Extending a gluing argument of Mrowka and Rollin we will show naturality of the contact invariant under strong symplectic cobordisms. Some applications include alternative proofs for: a) the vanishing of the contact invariant for overtwisted contact structures, b) its non-vanishing for the case of strongly fillable ones, c) strong fillings of planar contact structures must be negative definite.

Haofei Fan
Title: Embedded Grid Homology
Abstract: Grid homology is the combinatorial version of link Floer homology, as defined by Manolescu, Ozsváth and Sarkar. We define a natural notion of grid diagram, embedded grid diagram, for pointed links in three-sphere. Using this notion, we will prove the naturality of grid homology. In this talk, we will also provide some potential applications of the grid naturality maps. (with M. Marengon (UCLA) and M. Wong (LSU), work in progress)

Surena Hozoori
Title: Blair's Conjecture and Contact Dynamics
Abstract: It turns out that we can naturally define Riemannian structures "compatible" with a contact manifold. It is not well understood how we can get contact topological information from the local properties of these metrics, the way it is standard practice in the smooth category. One of the motivating conjectures along these lines has been Blair's conjecture, stating that there is no non-flat contact manifold of non-positive curvature. Using the theory of contact dynamics, we will give a new proof for the overtwisted case, which improves similar result by Etnyre-Komendraczyk-Massot.

Moses Koppendrayer
Title: Quasipositive Surfaces and Convex Surface Theory
Abstract: Quasipositive surfaces have been studied through braid constructions, open book decompositions and contact topology. In this talk I'll discuss some aspects of the last viewpoint particularly using convex surface theory, as well as presenting an example of its usefulness in a Lens Space.

Justin Lanier
Title: Adding points to configurations in closed balls
Abstract: Is there a continuous rule for adding a new distinct point to configurations of n distinct points in a closed ball? When n is 1, the answer is no by Brouwer’s fixed-point theorem. We answer the question in all other cases. This is joint work with Lei Chen and Nir Gadish.

Sangjin Lee
Title: Towards a Higher-dimensional construction of stable/unstable Lagrangian laminations
Abstract: A pseudo-Anosov surface automorphism has a pair of transverse foliations. Dimitrov, Haiden, Katzarkov and Kontsevich showed that a pair of transverse foliations in a surface induces a Bridgeland stability condition on the Fukaya category of the surface equipped with an area form. Also they asked the existence of higher-dimensional analogy of pseudo-Anosov surface automorphisms. In this talk, I give some symplectic automorphisms having a stable Lagrangian lamination, as candidates of pseudo-Anosov symplectic automorphisms.

Marissa Kawehi Loving
Title: Length spectra of q-differential metrics
Abstract: When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? For example, a flat metric on the torus can be determined by the lengths of only 3 distinct primitive closed curves. In this talk, we will explore which curves' lengths are needed to determine certain families of flat metrics on closed surfaces of genus at least 2.

William Olsen
Title: Trisecting Ozsv\'ath-Szab\'o four-manifold invariants
Abstract: Given a suitable decomposition of a smooth four-manifold, Ozsv\'ath and Szab\'o construct powerful invariants via a TQFT strategy known as Heegaard Floer homology. To compute these invariants for a given four-manifold, one must first calculate the relative invariants associated to each piece in the decomposition. In this brief talk, we'll demonstrate how one can use diagrams produced from the new theory of \emph{trisected Morse $2$-functions}, introduced by Gay and Kirby in 2012, to compute the relative invariants associated to each of the pieces.

Stephen McKean
Title: Enriching Bézout’s Theorem
Abstract: Bézout’s Theorem is a classical result from enumerative geometry that counts the number of intersections of projective planar curves over an algebraically closed field. Using a few tools from A1-homotopy theory, we enrich Bézout’s Theorem for perfect fields. Over non-algebraically closed fields, this enrichment imposes a relation on the tangent directions of the curves at their intersection points.

Maggie Miller
Title: Fibering ribbon disk complements
Abstract: Casson and Gordon showed that if K is a fibered ribbon knot, then K bounds a disk in a homotopy 4-ball whose complement is fibered by handlebodies. Recently, I proved that if D is a ribbon disk in the 4-ball satisfying some transversality condition (in particular, it is sufficient for D to have two minima), then D is similarly fibered. The proof involves constructing movies of singular fibrations on 3-manifolds. In five minutes, I'll show a few basic such movies.

Hyunki Min
Title: Contact structures on hyperbolic manifolds
Abstract: There are two basic questions in contact topology: Which manifolds admit tight contact structures, and on those that do, can we classify tight contact structures? In this talk, we present the first such classification on an infinite family of hyperbolic 3-manifolds. This is a joint work with James Conway.

Sinem Onaran
Title: Cryptographic applications of braids
Abstract: This talk gives an overview for braid based cryptosystems.

Joshua Pankau
Title: Constructing surfaces from positive integer matrices
Abstract: In this talk I will discuss a method of constructing a closed orientable surface from a positive integer matrix A, which will be the intersection matrix of a pair of filling multicurves on the surface. I will cover some properties of this construction, as well as give a bound on the genus of the surface that only depends on the size of the matrix. I will then discuss an application of this construction related to finding stretch factors of pseudo-Anosov maps coming from a construction due to Thurston.

JungHwan Park
Title: Concordance of knots in 3-manifolds.
Abstract: We investigate the disparity between smooth, topological, and piecewise linear concordance of knots in general 3-manifolds. This is joint work with Matthias Nagel, Patrick Orson, and Mark Powell.

Jacob Russell
Title: The Geometry of the Separating Curve Graph
Abstract: Topologists have long used complexes built from curves to study surfaces and their homeomorphisms. Since Masur and Minsky's celebrated proof of the hyperbolicity of the curve graph, an active industry has developed around understanding the geometry of these complexes. We build on the work of Vokes to study the geometry of the separating curve graph and prove that this graph is relatively hyperbolic in all the cases in which it is not hyperbolic.

Title: Link homology, bridge trisections and knotted surface invariants
Abstract: I'll describe an invariant of knotted surfaces in S4 defined via link homology and Meier-Zupan's bridge trisections.

Steve Trettel
Title: Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions
Abstract: The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations. Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories. Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual. This connects the aforementioned deformation theories and provides geometric motivation for a representation-theoretic observation of Cooper, Long, and Thistlethwaite.

Linh Truong
Title: An infinite rank summand of the homology cobordism group
Abstract: I will discuss a recent result on the homology cobordism group of integral homology three-spheres. It was previously known that the homology cobordism group admits a $\mathbb{Z}$ summand and a $\mathbb{Z}^\infty$ subgroup. We show that it admits a $\mathbb{Z}^\infty$ summand. The proof uses an algebraic modification of the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is inspired by an analogous argument in the setting of knot concordance due to Hom. This is joint work with Dai, Hom and Stoffregen.

Yvon Verberne
Title: Shadows from the pure mapping class group to the curve graph
Abstract: In Masur and Minsky's work on the geometry of the mapping class group, they construct special hierarchy paths connecting any pair of elements. Prominently, the shadow of a hierarchy path to the curve graph can always be re-parameterized to be a uniform quasi-geodesic. It seems intuitive that any geodesic in the mapping class group should also have this property. However, we construct a family of geodesics in the pure mapping class group of the five-times punctured sphere whose shadows to the curve graph cannot be re-parameterized to uniform quasi-geodesics. This is joint work with Kasra Rafi.

Melissa Zhang
Title: Symmetry and Localization
Abstract: I'll discuss the significance of localization in low-dimensional topology and exhibit examples and techniques from Floer and Khovanov homologies, including my joint work with Matthew Stoffregen on localization for the Khovanov homology of periodic links.

organizers: J. Etnyre, J. Hom, K. Kordek, P. Lambert-Cole, C. Leverson, D. Margalit, J. Park, and B. Strenner
Supported by the NSF and the Georgia Institute of Technology