



3:304:00 

Tea 


4:004:50 

Colloquium: Dale Rolfsen 

Ordered groups and ndimensional dynamics
Slides 
5:055:50 

Lightning talk session 1 

8 talks see below
Slides

6:00 

Social Gathering 

Publik Draft House





9:009:30 

Light refreshments 


9:3010:20 

Talk: Lisa Piccirillo 

Knot traces and spinelessness Scanned Notes 
10:2011:00 

Tea Break 


11:0011:50 

Talk: Siddhi Krishna 

Taut Foliations and the LSpace Conjecture Scanned Notes 
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: JungHwan Park 

Isotopy and equivalence of knots in 3manifolds Scanned Notes 
2:504:10 

Extended Tea Break 


4:105:00 

Talk: Johanna Mangahas 

Normal subgroups of mapping class groups

6:00 

Banquet 

At: South City Kitchen





9:009:30 

Light refreshments 


9:3010:20 

Talk: Nancy Scherich 

The Braid Groups and Their Representations Slides

10:2511:25 

Lighting talk session 3 

8 talks see below
Slides

11:2511:40 

Tea Break 


11:4012:30 

Talk: Daniel Ruberman 

Spines for spineless 4manifolds Slides 
All talks are in Skiles 006.
 Sudipta Kolay  Braid index of knotted surfaces 
 C.M. Michael Wong  Ribbon homology cobordisms 
 Lily Li & Caleb Partin  Finite Quotients of Braid Groups 
 Thomas Ng  Constructing free subgroups in nonpositive curvature 
 Michael Dougherty  Intrinsic Combinatorics for the Space of Generic Complex Polynomials 
 Kevin Kordek  Homomorphisms between braid groups 
 Hyun Ki Min  Cabling Legendrian knots 
 Steve Trettel  Raymarching the Thurston Geometries 
 Justin Lanier  Constraining mapping class group homomorphisms using finite subgroups 
 Anthony Sanchez  Distribution of slope gaps for slit tori 
 Elizabeth Field  Trees, dendrites, and the CannonThurston map 
 Christopher Anderson  An algorithm for an upper bound on splitting genus 
 Gage Martin  Annular Rasmussen invariants: Properties and 3braid classification 
 Rylee Lyman  Recognizing pseudoAnosov braids in Out(W_n) 
 Yvon Verberne  Constructing pseudoAnosov homeomorphisms using positive Dehn twists 
 Jose Roman Aranda Cuevas  Diagrams of $\star$trisections 
 Sunrose Shrestha  Statistics of random squaretiled surfaces 
 Ashlee Kalauli  The Word Problem for $\textsc{Art}\left(\widetilde{A_2}\right)$

 Peter Johnson  Small Seifert Fibered Zero Surgery 
 Kursat Yilmaz  Tight Contact Structures on Brieskorn Homology Spheres Sigma (2,3,6n+1) 
 Ian Montague  Tangle Invariants via Cornered Sutured Floer Homology 
 Sanghoon Kwak  On Translation Lengths of Anosov Maps on Curve Graph of Torus 
 Charles Stine  Relative Kirby Diagrams and Casson Tower Factories (joint with Bob Gompf) 
 Emily Shinkle  Finite Rigid Sets in the Arc Complex 
Titles and Abstracts
Taut Foliations and the LSpace Conjecture
The LSpace Conjecture is taking the lowdimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3manifold Y. In particular, it predicts a 3manifold Y isn't "simple" from the perspective of HeegaardFloer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, I'll discuss how to approach the converse  how we can try to identify and construct taut foliations in manifolds with "extra" Heegaard Floer homology. No background in Heegaard Floer homology or foliation theory will be assumed.
Normal subgroups of mapping class groups
Within a surface mapping class group, the wellknown normal subgroups (nearly all the ones with names) can be considered "geometric" in that their only automorphisms come from mapping classes, as proved by BrendleMargalit and McLeay. Free normal subgroups represent the opposite extreme, and these abound, by the result of Dahmani, Guirardel, and Osin on the normal closures of many pseudoAnosov mapping classes. I'll discuss new examples that include the middle of these two extremes, being normal subgroups isomorphic to rightangled Artin groups. This is joint work with Matt Clay and Dan Margalit.
Isotopy and equivalence of knots in 3manifolds
It is a wellknown and often used fact that the notions of (ambient) isotopy and equivalence coincide for knots in S^3, since any orientationpreserving homeomorphism of S^3 is isotopic to the identity. We compare the notions of equivalence and isotopy for knots in more general 3manifolds. We show that any orientationpreserving homeomorphism of a prime, oriented 3manifold which preserves free homotopy classes is isotopic to the identity, except in the single case of the Gluck twist acting on S^1xS^2. We give infinitely many examples of knots in S^1xS^2 whose isotopy classes are changed by a Gluck twist. This is joint work with Paolo Aceto, Corey Bregman, Christopher Davis, and Arunima Ray.
Knot traces and spinelessness
It remains at the forefront of 4manifold topology to construct simple closed 4manifolds with distinct smooth structures. Towards that end, it is of interest to construct simple 4manifolds with boundary with very distinct smooth structures. We produce infinitely many pairs of homeomorphic 4manifolds W and X homotopy equivalent to the 2sphere which have smooth structures distinguished by several formal properties: X is diffeomorphic to a knot trace and W is not, X admits a smooth spine and W does not admit a piecewise linear spine, X is geometrically simply connected and W is not, and W is a Stein domain and X is not. In particular our W are simple spineless 4manifolds, giving an alternative to Levine and Lidman’s recent solution to Problem 4.25 on Kirby’s list.
Ordered groups and ndimensional dynamics
A group is said to be torsionfree if it has no elements of finite order. An example is the group, under composition, of selfhomeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}. In fact this group has the stronger property of being leftorderable, meaning that the elements of the group can be ordered in a way that is invariant under leftmultiplication. If one restricts to piecewiselinear (PL) homeomorphisms, there exists a twosided (bi)ordering, an even stronger property of groups.
I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary. In the PL category, this group is leftorderable, but not biorderable, for all n>1. Also I will report on recent work of James Hyde showing that leftorderability fails for n>1 in the topological category.
Spines for spineless 4manifolds
A recent paper of LevineLidman gives examples of simply connected 4manifolds W, homotopy equivalent to a 2sphere, that do not admit PL spines. In other words, there is no piecewise linear (not necessarily locally flat) embedded sphere S that is a strong deformation retract of W. I will show that a family of the LevineLidman examples admit a topological spine—a locally PL sphere that is a strong deformation retract of W.
This is joint work with Hee Jung Kim.
The Braid Groups and Their Representations
Braid group theory is an interesting and versatile subject with applications in many different fields of mathematics including algebra, topology, and quantum computation. In this talk, I will give an introduction to the braid groups and share my intuition for why and how these groups are used. In particular, I will discuss the representations of the braid groups and some of the motivating open questions that fuel my research. Many of the famous representations of the braid groups are parametrized by a variable $q$ (these representations secretly come from quantum groups). I will share some of my results about choosing careful specializations of $q$ with the aim of structural results about the image of the representation.
Lightning talk
An algorithm for an upper bound on splitting genus
Unlike the Alexander polynomial of a knot, which can never be 0, the multivariable Alexander polynomial of link in $S^3$ can vanish. It is a classical result that the Alexander polynomial of a two component link $L\subset S^3$ with exterior $X$ vanishes exactly when the integral second homology of the universal abelian cover $\widetilde{X}$ is free on one generator as a module over the group ring $\mathbb{Z}[H_1(X)]$. We will discuss an algorithm for constructing a surface in $\widetilde{X}$ that realizes a generating homology class.
Diagrams of $\star$trisections
A trisection of a smooth connected 4manifold is a decomposition into three standard pieces. Similar to the case of Heegaard splittings in dimension three, a trisection is described by a trisection diagram: three sets of curves in a surface satisfying some properties. In general, it is not evident whether two trisection diagrams represent the same decomposition or even what 4manifold they depict.
In this talk I will explain how to soften the definition of a trisection in order to prove that a large family of genus three trisections is standard. This is joint work with Jesse Moeller.
Intrinsic Combinatorics for the Space of Generic Complex Polynomials
The space of degreen complex polynomials with distinct roots appears frequently and naturally throughout mathematics, but its shape and structure could be better understood. In recent and ongoing joint work with Jon McCammond, we present a deformation retraction of this space onto a simplicial complex associated to the braid group. In this talk, I will describe this retraction and discuss connections with the ideas of Bill Thurston and his collaborators.
Trees, dendrites, and the CannonThurston map
Given a short exact sequence of hyperbolic groups, $1\to H\to G\to Q\to 1$, Mahan Mitra has shown the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$ called the CannonThurston map. To every point $z$ in the Gromov boundary of $Q$, Mitra associates an â€œending laminationâ€ on H consisting of certain pairs of points in the boundary of $H$ identified by the CannonThurston map. We prove that for each such $z$, the quotient of the boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite: a treelike topological space.
Small Seifert Fibered Zero Surgery
Ichihara, Motegi, and Song discovered an infinite family of hyperbolic knots in $S^3$ such that zero surgery on them yield small Seifert fiber spaces. On the other hand, Hedden, Kim, Mark, and Park found an infinite family of small Seifert fiber spaces with $b_{1} = 1$ and weight $1$ fundamental group which are not obtained via zero surgery on a knot in $S^3$. In this talk, we discuss work towards producing more examples of small Seifert fibered spaces which are obtained by zero surgery on a knot in $S^3$, as well as strategies for finding obstructions.
The Word Problem for $\textsc{Art}\left(\widetilde{A_2}\right)$
In general, the Word Problem for Artin groups remains open. In this lightning talk we discuss how a particular solution to the word problem for $\textsc{Art}\left(\widetilde{A_2}\right)$ involving an infinite generating set will lead to a solution on its classical generating set.
Braid index of knotted surfaces
Braided surfaces play a similar role in understanding the knotted surfaces in four space, as classical braids for knots in three space. In this talk, I will introduce the notion of braided surfaces, and discuss properties of braid index of knotted surfaces and contrast it with that of classical knots.
Homomorphisms between braid groups
I will discuss some recent work, joint with Chen and Margalit, on the classification of homomorphisms between braid groups. The main result is a complete classification of all homomorphisms from the braid group on $n$ strands to the braid group on $2n$ strands when $n$ is at least 5.
On Translation Lengths of Anosov Maps on Curve Graph of Torus
There has been much research to estimate stable translation length for nonsporadic surfaces. To give an instance, MasurMinsky showed a pseudoAnosov map has a quasigeodesic axis on the curve graph of the nonsporadic surface, and its stable translation length is positive, which is strengthened to be rational later by Bowditch. However, to the best of our knowledge, there was no literature dealing with the same question for sporadic surfaces. I will discuss joint work with Hyungryul Baik, Changsub Kim, and Hyunshik Shin in which an Anosov map has a geodesic axis on the curve graph of a torus.
Constraining mapping class group homomorphisms using finite subgroups
We give a new short proof of a theorem of AramayonaSouto that constrains homomorphisms between mapping class groups of closed surfaces. The proof proceeds by analyzing finite subgroups. This is joint work with Lei Chen.
Recognizing pseudoAnosov braids in Out($W_n$)
Bestvinaâ€“Handel showed that pseudoAnosov homeomorphisms of surfaces with one boundary component are in bijective correspondence with train track maps of graphs for fully irreducible outer automorphisms preserving some nontrivial conjugacy class. Recently I gave a construction for (relative) train track maps of orbigraphs for outer automorphisms of free products. In the case of $W_n$, the free product of $n$ groups of order 2, I will show that fully irreducible outer automorphisms preserving some nontrivial conjugacy class are in bijective correspondence with pseudoAnosov braids on oncepunctured orbifolds. Interestingly the statement remains "morally true" for more general free products, but there is some subtlety.
Finite Quotients of Braid Groups
A natural question to ask when studying groups is what the smallest nontrivial finite quotient of this group is. Further, what if we ask about the smallest finite, noncyclic quotient? We will discuss joint work with Alice Chudnovsky and Kevin Kordek, in which we derive a lower bound on the size of finite groups the braid group can map to noncyclically. Moreover, we will discuss totally symmetric sets, a recent idea developed by Dan Margalit and Kevin Kordek, and the primary tool used to achieve the bound.
Annular Rasmussen invariants: Properties and 3braid classification
Annular Rasmussen invariants are invariants of annular links which generalize the Rasmussen s invariant and come from a $\mathbb{Z} \oplus \mathbb{Z}$ filtration on KhovanovLee homology. In this talk we will explain some reasons to care about the annular Rasmussen invariants. Additionally we will state theorems about restrictions on the possible values of annular Rasmussen invariants and a computation of the invariants for all 3braid closures, or conjugacy classes of 3braids, phrased with respect to Murasugi's classification of 3braids.
Cabling Legendrian knots
If the classification of Legendrian knots of a given knot type is known, is it possible to classify Legendrian knots of its cables? Etnyre and Honda showed that if a given knot type is Legendrian simple and uniformly thick, then its cables are also Legendrian simple. Tosun improved this result for positive cables by removing the uniform thickness condition. In this talk, we will give a complete classification of Legendrian positive cables. This is a joint work with Apratim Chakraborty and John Etnyre.
Tangle Invariants via Cornered Sutured Floer Homology
As of date there are myriad tangle invariants for tangles in $S^{2}\times I$ and $B^{3}$ which refine link Floer homology; given a link $L$ in $S^{3}$ and a tangle decomposition $L = T_{1}\coprod \cdots\coprod T_{n}$ obtained by slicing a link diagram along horizontal crosssections, one can recover the link Floer homology of $L$ by gluing the tangle invariants associated to $\{T_{i}\}$ from
"toptobottom", making use of \textit{bordered} Heegaard Floer homology. In this talk, we outline the construction of an invariant which, given a tangle decomposition of link $L$ obtained by slicing a link diagram along both horizontal \textit{and} vertical crosssections, recovers $HFL^{}(L,S^{3})$ by gluing the tangle invariants of the component pieces from "toptobottom" \textit{and} "sidetoside", using an extension of DouglasLipshitzManolescu's \textit{cornered} Heegaard Floer homology to the setting of sutured manifolds.
Constructing free subgroups in nonpositive curvature
The strong Tits alternative holds for groups whose subgroups are either virtually abelian or contain a free subgroup. Many groups that arise naturally in studying nonpositively curved manifolds, including fundamental groups of closed hyperbolic manifolds, cubical groups, and mapping class groups, satisfy the strong Tits alternative.Â Verifying when pairs of elements generate a free subgroup is wellknown to be quite challenging. In this talk I will describe joint work with Radhika Gupta and Kasia Jankiewicz explicitly constructing of free semigroup bases that rely on combinatorial characterizations of the dynamics of group actions on cube complexes and applications to uniform exponential growth.
Distribution of slope gaps for slit tori
Consider the class of translation surfaces given by gluing two identical tori along a slit. Every such surface has genus two and two conetype singularities of angle $4\pi$. There is a distinguished set of geodesics called saddle connections that are the geodesics between cone points. We can recover a vector in the plane representing the saddle connection by keeping track of the amount that the saddle connection moves in the vertical and horizontal direction. How random is the set of saddle connections? We shed light to this question by considering the gap distribution of slopes of saddle connections.
Finite Rigid Sets in the Arc Complex
The arc complex of a surface is a simplicial complex whose vertices correspond to isotopy classes of arcs and whose higher dimensional simplices correspond to collections of arcs with disjoint representatives. In 2010, Irmak and McCarthy proved that every automorphism of the arc complex is induced by a homeomorphism of the surface. I strengthen this by finding finite sets in the arc complex whose embeddings are always induced by homeomorphisms.
Statistics of random squaretiled surfaces
Squaretiled surfaces are finite branched covers of the standard square torus, with branching over exactly one point. These can be also thought of as surfaces obtained by taking finitely many Euclidean unit squares, and gluing the edges in parallel pairs. In this talk we will see topological and geometric statistics of random squaretiled surfaces using a simple combinatorial model.
Relative Kirby Diagrams and Casson Tower Factories (joint with Bob Gompf)
Casson handles, which are homeomorphic but not diffeomorphic to standard, 4dimensional 2handles are a central ingredient in the proof of Freedman’s hcobordism
theorem. The process which constructs Casson handles can, a priori, construct uncountably many which may or may not be smoothly distinct. These are indexed by certain signed and
weighted trees. Our project aims to generalize 4dimensional Kirby Calculus slightly to obtain clearer pictures of how Casson handles are built and embedded into other 4manifolds, and then to construct more invariantfriendly pairs of manifolds, each containing a Casson handle, whose smooth equivalence is determined by the smooth equivalence of the respective Casson handles.
Raymarching the Thurston Geometries
We see objects in the world around us through light that bounces off them and into our eyes.Â As such, sight is a kind of geometrical sense, giving us a way of probing space with a spray of geodesics through each eye.Â This interpretation has been used to great effect in computer graphics, where an image is rendered via shooting out rays from each pixel into an imagined scene and computing its interactions / reflections with CG objects.Â Â
We generalize this marching along light rays to marching along geodesics in Riemannian homogeneous spaces, allowing us to "see inside" other 3dimensional geometries, such as hyperbolic space, S2xR and Nil.Â This project is joint work with Henry Segerman, Sabetta Matsumoto, Remi Coulon and Brian Day.
Constructing pseudoAnosov homeomorphisms using positive Dehn twists
Thurston obtained the first construction of pseudoAnosov homeomorphisms by showing the product of two parabolic elements is pseudoAnosov. A related construction by Penner involved constructing whole semigroups of pseudoAnosov homeomorphisms by taking appropriate compositions of Dehn twists along certain families of curves. In this talk, I will present a new construction of pseudoAnosov homeomorphisms using only positive Dehn twists.
Ribbon homology cobordisms
A cobordism between 3manifolds is ribbon if it has no 3handles. Such cobordisms arise naturally from several different contexts. In this talk, we describe a few obstructions to their existence, from Heegaard and instanton Floer homologies, character varieties, and Thurston geometries, and some applications.
Tight Contact Structures on Brieskorn Homology Spheres Sigma (2,3,6n+1)
Brieskorn homology spheres are special kind of Small Seifert fibered spaces which are the Seifert fibered spaces over 2sphere with exactly three singular fibers. Mark and Tosun showed that this family admits exactly two tight contact structures. The proof involves Kirby calculus, convex surface theory and Honda's bypass technique. In five minutes, I will construct the Stein pictures of this two contact structures and give a basic idea that how did they find the upper bound for the number of tight contact structures.
