All talks are in Skiles 006.

Lightning talks session 1:

Lightning talks session 2:

Lightning talks session 3:

Titles and Abstracts

Siddhi Krishna
Title: Taut Foliations and the L-Space Conjecture
Abstract: The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer 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.

Johanna Mangahas
Title: Normal subgroups of mapping class groups
Abstract: Within a surface mapping class group, the well-known normal subgroups (nearly all the ones with names) can be considered "geometric" in that their only automorphisms come from mapping classes, as proved by Brendle--Margalit 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 pseudo-Anosov mapping classes. I'll discuss new examples that include the middle of these two extremes, being normal subgroups isomorphic to right-angled Artin groups. This is joint work with Matt Clay and Dan Margalit.

JungHwan Park
Title: Isotopy and equivalence of knots in 3-manifolds
Abstract: It is a well-known and often used fact that the notions of (ambient) isotopy and equivalence coincide for knots in S^3, since any orientation-preserving homeomorphism of S^3 is isotopic to the identity. We compare the notions of equivalence and isotopy for knots in more general 3-manifolds. We show that any orientation-preserving homeomorphism of a prime, oriented 3-manifold 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.

Lisa Piccirillo
Title: Knot traces and spinelessness
Abstract: It remains at the forefront of 4-manifold topology to construct simple closed 4-manifolds with distinct smooth structures. Towards that end, it is of interest to construct simple 4-manifolds with boundary with very distinct smooth structures. We produce infinitely many pairs of homeomorphic 4-manifolds W and X homotopy equivalent to the 2-sphere 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 4-manifolds, giving an alternative to Levine and Lidman’s recent solution to Problem 4.25 on Kirby’s list.

Dale Rolfsen
Title: Ordered groups and n-dimensional dynamics
Abstract: A group is said to be torsion-free if it has no elements of finite order. An example is the group, under composition, of self-homeomorphisms (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 left-orderable, meaning that the elements of the group can be ordered in a way that is invariant under left-multiplication. If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (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 left-orderable, but not bi-orderable, for all n>1. Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.

Daniel Ruberman
Title: Spines for spineless 4-manifolds
Abstract: A recent paper of Levine-Lidman gives examples of simply connected 4-manifolds W, homotopy equivalent to a 2-sphere, 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 Levine-Lidman 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.

Nancy Scherich
Title: The Braid Groups and Their Representations
Abstract: 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.

Christopher Anderson
Title: An algorithm for an upper bound on splitting genus
Abstract: 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.

Jose Roman Aranda Cuevas
Title: Diagrams of $\star$-trisections
Abstract: A trisection of a smooth connected 4-manifold 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 4-manifold 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.

Michael Dougherty
Title: Intrinsic Combinatorics for the Space of Generic Complex Polynomials
Abstract: The space of degree-n 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.

Elizabeth Field
Title: Trees, dendrites, and the Cannon-Thurston map
Abstract: 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 Cannon-Thurston map. To every point $z$ in the Gromov boundary of $Q$, Mitra associates an “ending lamination” on H consisting of certain pairs of points in the boundary of $H$ identified by the Cannon-Thurston 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 tree-like topological space.

Peter Johnson
Title: Small Seifert Fibered Zero Surgery
Abstract: 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.

Ashlee Kalauli
Title: The Word Problem for $\textsc{Art}\left(\widetilde{A_2}\right)$
Abstract: 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.

Sudipta Kolay
Title: Braid index of knotted surfaces
Abstract: 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.

Kevin Kordek
Title: Homomorphisms between braid groups
Abstract: 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.

Sanghoon Kwak
Title: On Translation Lengths of Anosov Maps on Curve Graph of Torus
Abstract: There has been much research to estimate stable translation length for non-sporadic surfaces. To give an instance, Masur-Minsky showed a pseudo-Anosov map has a quasi-geodesic axis on the curve graph of the non-sporadic 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.

Justin Lanier
Title: Constraining mapping class group homomorphisms using finite subgroups
Abstract: We give a new short proof of a theorem of Aramayona--Souto that constrains homomorphisms between mapping class groups of closed surfaces. The proof proceeds by analyzing finite subgroups. This is joint work with Lei Chen.

Rylee Lyman
Title: Recognizing pseudo-Anosov braids in Out($W_n$)
Abstract: Bestvina–Handel showed that pseudo-Anosov 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 pseudo-Anosov braids on once-punctured orbifolds. Interestingly the statement remains "morally true" for more general free products, but there is some subtlety.

Lily Li & Caleb Partin
Title: Finite Quotients of Braid Groups
Abstract: A natural question to ask when studying groups is what the smallest non-trivial finite quotient of this group is. Further, what if we ask about the smallest finite, non-cyclic 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 non-cyclically. 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.

Gage Martin
Title: Annular Rasmussen invariants: Properties and 3-braid classification
Abstract: 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 Khovanov-Lee 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 3-braid closures, or conjugacy classes of 3-braids, phrased with respect to Murasugi's classification of 3-braids.

Hyun Ki Min
Title: Cabling Legendrian knots
Abstract: 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.

Ian Montague
Title: Tangle Invariants via Cornered Sutured Floer Homology
Abstract: 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 cross-sections, one can recover the link Floer homology of $L$ by gluing the tangle invariants associated to $\{T_{i}\}$ from "top-to-bottom", 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 cross-sections, recovers $HFL^{-}(L,S^{3})$ by gluing the tangle invariants of the component pieces from "top-to-bottom" \textit{and} "side-to-side", using an extension of Douglas-Lipshitz-Manolescu's \textit{cornered} Heegaard Floer homology to the setting of sutured manifolds.

Thomas Ng
Title: Constructing free subgroups in nonpositive curvature
Abstract: 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.  Verifying when pairs of elements generate a free subgroup is well-known 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.

Anthony Sanchez
Title: Distribution of slope gaps for slit tori
Abstract: Consider the class of translation surfaces given by gluing two identical tori along a slit. Every such surface has genus two and two cone-type 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.

Emily Shinkle
Title: Finite Rigid Sets in the Arc Complex
Abstract: 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.

Sunrose Shrestha
Title: Statistics of random square-tiled surfaces
Abstract: Square-tiled 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 square-tiled surfaces using a simple combinatorial model.

Charles Stine
Title: Relative Kirby Diagrams and Casson Tower Factories (joint with Bob Gompf)
Abstract: Casson handles, which are homeomorphic but not diffeomorphic to standard, 4-dimensional 2-handles are a central ingredient in the proof of Freedman’s h-cobordism 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 4-dimensional Kirby Calculus slightly to obtain clearer pictures of how Casson handles are built and embedded into other 4-manifolds, and then to construct more invariant-friendly pairs of manifolds, each containing a Casson handle, whose smooth equivalence is determined by the smooth equivalence of the respective Casson handles.

Steve Trettel
Title: Raymarching the Thurston Geometries
Abstract: We see objects in the world around us through light that bounces off them and into our eyes.  As such, sight is a kind of geometrical sense, giving us a way of probing space with a spray of geodesics through each eye.  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.   We generalize this marching along light rays to marching along geodesics in Riemannian homogeneous spaces, allowing us to "see inside" other 3-dimensional geometries, such as hyperbolic space, S2xR and Nil.  This project is joint work with Henry Segerman, Sabetta Matsumoto, Remi Coulon and Brian Day.

Yvon Verberne
Title: Constructing pseudo-Anosov homeomorphisms using positive Dehn twists
Abstract: Thurston obtained the first construction of pseudo-Anosov homeomorphisms by showing the product of two parabolic elements is pseudo-Anosov. A related construction by Penner involved constructing whole semigroups of pseudo-Anosov homeomorphisms by taking appropriate compositions of Dehn twists along certain families of curves. In this talk, I will present a new construction of pseudo-Anosov homeomorphisms using only positive Dehn twists.

C.M. Michael Wong
Title: Ribbon homology cobordisms
Abstract: A cobordism between 3-manifolds is ribbon if it has no 3-handles. 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.

Kursat Yilmaz
Title: Tight Contact Structures on Brieskorn Homology Spheres Sigma (2,3,6n+1)
Abstract: Brieskorn homology spheres are special kind of Small Seifert fibered spaces which are the Seifert fibered spaces over 2-sphere 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.