



3:304:00 

Tea 


4:004:50 

Talk:Tara Brendle* 

Congruence subgroups of braid groups 
5:055:50 

Lightning talk session 1 

7 talks see below
Slides

6:00 

Social Gathering 

Publik Draft House





9:009:30 

Light refreshments 


9:3010:20 

Talk: Andrew Putman 

The stable cohomology of the moduli space of curves with level structures 
10:2011:00 

Tea Break 


11:0011:50 

Talk: Kevin Kordek 

The rational cohomology of the level 4 braid group 
11:5012:30 

Catered Lunch 

Fox Bros BBQ 
12:301:30 

Lighting talk session 2 

8 talks see below
Slides

1:302:00 

break 


2:002:50 

Talk: Sucharit Sarkar 

Khovanov homotopy types Scanned Notes 
2:504:10 

Extended Tea Break 


4:105:00 

Talk: Allison Miller 

Satellite operators on concordance
Slides 
6:00 

Banquet 

At: South City Kitchen





9:009:30 

Light refreshments 


9:3010:20 

Talk: Juanita Pinzon Caicedo 

Satellites of Infinite Rank in Concordance Scanned Notes

10:2511:25 

Lighting talk session 3 

8 talks see below
Slides

11:2511:40 

Tea Break 


11:4012:30 

Talk: Gordana Matic 

Spectral order contact invariant Scanned Notes 
All talks are in Skiles 006.
* This is a colloquium talk
 JungHwan Park  Concordance of knots in 3manifolds 
 Marissa Kawehi Loving  Length spectra of qdifferential 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 2component links with multivariable Alexander polynomial 0 
 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 Higherdimensional 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 
 Steve Trettel  Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions 
 Melissa Zhang  Symmetry and Localization 
 Haofei Fan  Embedded Grid Homology 
 William Olsen  Trisecting OzsvathSzabo fourmanifold invariants 
 Mariano Echeverria  Naturality of the Contact Invariant in Monopole Floer Homology under Strong Symplectic Cobordisms 
 Sahana H Balasubramanya  Acylindrical group actions on quasitrees 
 Joshua Pankau  Constructing surfaces from positive integer matrices 
 Yvon Verberne  Shadows from the pure mapping class group to the curve graph 
Titles and Abstracts
Congruence subgroups of braid groups
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.
Two theorems about braid groups
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.
Spectral order contact invariant
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, nondecreasing 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 3manifolds. We exhibit an infinite family of examples with vanishing Heegaard Floer contact invariant on which our invariant assumes an unbounded sequence of finite and nonzero values.
This is joint work with \c Ca\u gatay Kutluhan, Jeremy Van HornMorris and Andy Wand.
Satellite operators on concordance
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 4genus; and there are several infinite filtrations of $\mathcal{C}$, most famously that of CochranOrrTeichner. Every pattern, or knot in a solid torus, induces a welldefined 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 satelliteinduced 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.
Satellites of Infinite Rank in Concordance
Oriented knots are said to be concordant if they cobound an embedded cylinder in the interval times the 3sphere. 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 4manifolds. Indeed, many questions pertaining to 4manifolds with small topology (like the 4sphere) 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.
The stable cohomology of the moduli space of curves with level structures
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
MillerMoritaMumford classes.
Khovanov homotopy types
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
Splitting surfaces of 2component links with multivariable Alexander polynomial 0
A 2component 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.
Acylindrical group actions on quasitrees
A group G is called acylindrically hyperbolic if it admits a nonelementary 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 nonelementary acylindrical action on a quasitree . A quasitree is a connected graph quasiisometric 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.
Orthoscheme Configuration Spaces
Configuration spaces of manifolds provide natural examples of highdimensional spaces with lowdimensional 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 nonpositively curved. In recent work with Jon McCammond and Stefan Witzel, we introduce a similar type of configuration space for directed graphs and, more generally, Deltacomplexes. In this talk, I will define orthoscheme configuration spaces and present what is known so far.
Naturality of the Contact Invariant in Monopole Floer Homology under Strong Symplectic Cobordisms
The contact invariant is an element in the monopole Floer homology groups of a three manifold associated to a contact structure. A nonvanishing 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 nonvanishing for the case of strongly fillable ones,
c) strong fillings of planar contact structures must be negative definite.
Embedded Grid Homology
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 threesphere. 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)
Blair's Conjecture and Contact Dynamics
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 nonflat contact manifold of nonpositive curvature. Using the theory of contact dynamics, we will give a new proof for the overtwisted case, which improves similar result by EtnyreKomendraczykMassot.
Quasipositive Surfaces and Convex Surface Theory
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.
Adding points to configurations in closed balls
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 fixedpoint theorem. We answer the question in all other cases. This is joint work with Lei Chen and Nir Gadish.
Towards a Higherdimensional construction of stable/unstable Lagrangian laminations
A pseudoAnosov 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 higherdimensional analogy of pseudoAnosov surface automorphisms. In this talk, I give some symplectic automorphisms having a stable Lagrangian lamination, as candidates of pseudoAnosov symplectic automorphisms.
Length spectra of qdifferential metrics
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.
Trisecting Ozsv\'athSzab\'o fourmanifold invariants
Given a suitable decomposition of a smooth fourmanifold, Ozsv\'ath and Szab\'o construct powerful invariants via a TQFT strategy known as Heegaard Floer homology. To compute these invariants for a given fourmanifold, 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.
Enriching Bézout’s Theorem
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 A1homotopy theory, we enrich Bézout’s Theorem for perfect fields. Over nonalgebraically closed fields, this enrichment imposes a relation on the tangent directions of the curves at their intersection points.
Fibering ribbon disk complements
Casson and Gordon showed that if K is a fibered ribbon knot, then K bounds a disk in a homotopy 4ball whose complement is fibered by handlebodies. Recently, I proved that if D is a ribbon disk in the 4ball 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 3manifolds. In five minutes, I'll show a few basic such movies.
Contact structures on hyperbolic manifolds
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 3manifolds. This is a joint work with James Conway.
Cryptographic applications of braids
This talk gives an overview for braid based cryptosystems.
Constructing surfaces from positive integer matrices
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 pseudoAnosov maps coming from a construction due to Thurston.
Concordance of knots in 3manifolds.
We investigate the disparity between smooth, topological, and piecewise linear concordance of knots in general 3manifolds. This is joint work with Matthias Nagel, Patrick Orson, and Mark Powell.
The Geometry of the Separating Curve Graph
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.
Link homology, bridge trisections and knotted surface invariants
I'll describe an invariant of knotted surfaces in S4 defined via link homology and MeierZupan's bridge trisections.
Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions
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 representationtheoretic observation of Cooper, Long, and Thistlethwaite.
An infinite rank summand of the homology cobordism group
I will discuss a recent result on the homology cobordism group of integral homology threespheres. 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 HendricksManolescu and HendricksManolescuZemke. 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.
Shadows from the pure mapping class group to the curve graph
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 reparameterized to be a uniform quasigeodesic. 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 fivetimes punctured sphere whose shadows to the curve graph cannot be reparameterized to uniform quasigeodesics. This is joint work with Kasra Rafi.
Symmetry and Localization
I'll discuss the significance of localization in lowdimensional 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.
