All talks are in Skiles 006.

*** Cancelled due to weather.

Lightning talks session 1:

Lightning talks session 2:

Lightning talks session 3:

Titles and Abstracts

Tori Akin
Title: Automorphisms of the Punctured Mapping Class Group
Abstract: We can describe the point-pushing subgroup of the mapping class group topologically as the set of maps that push a puncture around loops in the surface. However, we can characterize this topological subgroup in purely algebraic terms. Using group theoretic tools and a classic theorem of Burnside, we can recover a result of Ivanov-McCarthy establishing the triviality of Out(Mod±). To this end, we’ll demonstrate that the point-pushing subgroup is “unique” in the mapping class group.

Lei Chen
Title: The number of fiberings of a surface bundle over a surface
Abstract: For a closed manifold M, let SFib(M) be the number of distinct fiberings of M as a fiber bundle with fiber a closed surface of genus>1. In this talk I will talk about the history of the problem of classifying number of fiberings and describe a criterion for a 4-manifold M to have SFib(M)=2. I will then apply the criterion to classify the number of fiberings of the famous Atiyah--Kodaira manifold and any finite cover of a trivial surface bundle.

Diana Hubbard
Title: The braid index, the fractional Dehn twist coefficient, and Upsilon
Abstract: The braid index of a knot is the least number of strands necessary to represent the knot as a closure of a braid. Informally, the fractional Dehn twist coefficient of a braid measures the amount of "full twisting" it has. In this talk I will discuss joint work with Peter Feller showing that if an n-braid has fractional Dehn twist coefficient greater than n-1 then its closure has braid index n. A crucial tool is a characterization of the fractional Dehn twist coefficient in terms of a slope of the homogenization of Upsilon, a knot concordance homomorphism defined by Ozsvath, Stipsicz, and Szabo.

Justin Lanier
Title: Three theorems about generating mapping class groups
Abstract: We will first show how mapping class groups can be generated by a small number of periodic mapping classes of any fixed nontrivial order. We will then discuss joint work with Dan Margalit that produces an abundance of mapping classes whose normal closure is the whole mapping class group. Two such families are the nontrivial periodic mapping classes (excepting the hyperelliptic involution) and the pseudo-Anosov mapping classes with sufficiently small stretch factor. Our pseudo-Anosov examples answer a question of Darren Long from 1986.

Tye Lidman
Title: Band surgeries on the trefoil
Abstract: Attaching a band to a knot in S^3 produces a cobordism to a new knot or link. This process can reveal significant information about the surfaces that knots can bound in B^4. We characterize certain band surgeries on the trefoil knot and solve a related problem about Dehn surgeries between lens spaces. This is joint work with Allison Moore and Mariel Vazquez.

Ciprian Manolescu
Title: A sheaf-theoretic model for SL(2,C) Floer homology
Abstract: I will explain the construction of a new homology theory for three-manifolds, defined using perverse sheaves on the SL(2,C) character variety. Our invariant is a model for an SL(2,C) version of Floer’s instanton homology. I will present a few explicit computations for Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. This is joint work with Mohammed Abouzaid.

Emmy Murphy
Title: Planar graphs and Legendrian surfaces
Abstract: Associated to a planar cubic graph, there is a closed surface in R^5, as defined by Treumann and Zaslow. R^5 has a canonical geometry, called a contact structure, which is compatible with the surface. The data of how this surface interacts with the geometry recovers interesting data about the graph, notably its chromatic polynomial. This also connects with pseudo-holomorphic curve counts which have boundary on the surface, and by looking at the resulting differential graded algebra coming from symplectic field theory, we obtain new definitions of n-colorings which are strongly non-linear as compared to other known definitions. There are also relationships with SL_2 gauge theory, mathematical physics, symplectic flexibility, and holomorphic contact geometry. During the talk we'll explain the basic ideas behind the various fields above, and why these various concepts connect.

Balázs Strenner
Title: Fibrations of 3-manifolds, tilings and nowhere continuous functions
Abstract: We start with a 3-manifold fibering over the circle and investigate how the pseudo-Anosov monodromies change as we vary the fibration. Fried proved that the stretch factor of the monodromies varies continuously (when normalized in the appropriate sense). In sharp contrast, we show that another numerical invariant, the asymptotic translation length in the arc complex, does not vary continuously no matter how it is normalized. We will show that the discontinuity of these functions is related to the (lack of) tilings of Euclidean spaces with certain shapes.

Bahar Acu
Title: Fillings of iterated planar contact manifolds
Abstract: Planar contact manifolds (contact manifolds whose supporting open book has planar pages) have been intensively studied to understand several aspects of three dimensional contact geometry. In this talk, we define iterated planar contact manifolds, an analog of this idea in higher dimensions, and explain some existing 3-dimensional results regarding symplectic fillings (cobordism from the empty set to a contact manifold) of planar contact manifolds and speculate some higher-dimensional preliminary results. This is a joint work in progress with J. Etnyre and B. Ozbagci.

Sarah Davis
Title: Covering spaces, mapping class groups, and the symplectic representation
Abstract: The mapping class group of a given surface acts on the first homology of that surface, and this action induces a representation into the group of symplectic matrices. In joint work with Stordy, Winarski, and Zhou, we study a 3-fold cyclic branched covering of a surface of genus 2 over a sphere. In particular, we determine the image of a certain subgroup of the mapping class, known as the symmetric mapping class group, in the symplectic group. Our result addresses a question of McMullen.

Michael Dougherty
Title: Braids with Boundary-Parallel Strands
Abstract: It is common in inductive arguments for the braid group to consider subgroups of braids which leave some of the strands fixed. Extending this idea, we can describe strands which “wrap around the outside” of the braid. The resulting sets of braids then have natural interpretations as subspaces of a simplicial complex on which the braid group acts geometrically. In this talk I describe joint work with Jon McCammond and Stefan Witzel which exhibits the structure of these complexes and implications on the curvature of the braid group.

Haofei Fan
Title: Unoriented cobordism maps on link Floer homology
Abstract: We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, bipartite disoriented link cobordism, which tracks not only the motion of the basepoints of the link but also the motion of index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a bipartite disoriented link cobordism. Furthermore, we give a comparison with Oszv\'{a}th-Stipsicz-Szab\'{o}’s and Manolescu’s constructions of link cobordism maps for an unoriented band move.

Sudipta Kolay
Title: Generalized Alexander's Theorem
Abstract: A classical theorem of Alexander about closed braids allows us to study knots in three-space using the theory of braids. In this talk, we will define closed braids in higher dimensions, and discuss generalizations of Alexander's theorem.

Kevin Kordek
Title: A large abelian quotient of the level 4 braid group
Abstract: It is generally a difficult problem to compute the Betti numbers of a given finite-index subgroup of an infinite group, even if the Betti numbers of the ambient group are known. In this talk, I will present a new (large) lower bound on the first Betti number of the level 4 braid group, which is the kernel of the mod 4 reduction of the integral Burau representation. This is joint work with Dan Margalit.

Ryan Leigon
Title: An Excursion in Gluing Maps
Abstract: Sutured Floer homology is a powerful 3-manifold invariant useful for studying links and contact structures. The theory is functorial: cobordisms between sutured manifolds give rise to maps on sutured Floer homology which factor through gluing maps defined by Honda, Kazez, and Matic. However, Honda-Kazez-Matic maps are difficult to compute, even in the simplest cases. We show that these maps can be described by Zarev gluing maps, which are readily computable in applications. This work is joint with Federico Salmoiraghi and recreates unpublished work of Zarev.

Marissa Loving
Title: Least dilatation of pure surface braids
Abstract: The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. For the n=1 case, much is known about this group including upper and lower bounds on the least dilatation of its pseudo-Anosovs due to Dowdall and Aougab—Taylor. I am interested in the least dilatation of pseudo-Anosov pure surface braids for n>1 punctures. I will describe the upper and lower bounds I have proved as a function of g and n.

Yu Pan
Title: Augmentations and immersed exact Lagrangian fillings
Abstract: It is well known that not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all the augmentations come from possibly immersed exact Lagrangian fillings. For a Legendrian knot in J^1(M), take an immersed exact Lagrangian filling that can be lifted to an embedded Legendrian L in J^1(R \times M). For any augmentation of L, we associate an induced augmentation of the Legendrian knot, whose homotopy class only depends on the compactly supported Legendrian isotopy type of L and the homotopy class of its augmentation. In addition, we also give a relation between the linearized contact homology of the Legendrian knot and some type of homology of L using wrapped Floer theory. This is a joint work with Dan Rutherford in progress.

Joshua Pankau
Title: Salem number stretch factors
Abstract: Associated to every pseudo-Anosov homeomorphism of a closed orientable surface is a real number called the stretch factor. Thurston showed that the stretch factor of any pseudo-Anosov map is an algebraic unit, but it is still an open question which algebraic units appear as stretch factors. In this talk, I will discuss my recent progress on this question where I focused on a construction of pseudo-Anosov maps due to Thurston, and a particular algebraic unit known as a Salem number.

Jonathan Paprocki
Title: Topological Quantum Compiling via Quantum Teichmuller Theory
Abstract: Topological quantum compiling is roughly the following problem: given an element of an algebra representation coming from a topological quantum field theory, find a "minimum complexity" element of the algebra in the preimage of a small neighborhood of the element. We present preliminary work towards an algorithm for this problem in the case of the quantum representations of Kauffmann bracket skein algebras of surfaces coming from quantum Teichmuller theory via the quantum trace map of Bonahon and Wong, which conjecturally has SU(2) Chern-Simons theory as a special case. The solution to this problem has applications in topological quantum computing.

Aamir Rasheed
Title: Essential embeddings and immersions of surfaces in a 3-manifold
Abstract: In this talk we will discuss a theorem which gives a sufficient condition for two embedded surfaces in an irreducible 3-manifold to be isotopic. We shall also discuss some generalizations, such as how to detect whether two essential immersions of surfaces are homotopic.

Adam Saltz
Title: Why are there so many spectral sequences from Khovanov homology?
Abstract: There are at least eight spectral sequences from Khovanov homology to other invariants of links. These invariants come from gauge theory, symplectic topology, and representation theory, so it's striking that they all are related to Khovanov homology. Baldwin, Hedden, and Lobb call these Khovanov-Floer theories and prove some cool facts about them using spectral sequences. I'll discuss techniques to prove stronger theorems by avoiding spectral sequences altogether.

Bakul Sathaye
Title: Link obstruction to Riemannian smoothings of a locally CAT(0) manifold.
Abstract: By work of Davis, Januszkiewicz and Lafont, there exist obstructions to having a smooth Riemannian metric of non-positive sectional curvature on a locally CAT(0) manifold. I will describe a new obstruction that comes from links in the boundary at infinity. The universal cover of such a manifold satisfies Hruska’s isolated flats condition and contains a collection of 2-dimensional flats with the property that their boundaries form a non-trivial link.

Jonathan Simone
Title: Towards a new construction of exotic 4-manifolds
Abstract: In this talk I will introduce a new construction (similar to the rational blow-down) that can potentially be used build exotic 4-manifolds. In particular, I will describe the machinery I have developed and potential examples of new exotic 4-manifolds. I will also describe a recent result that classifies tight contact structures with no Giroux torsion on plumbed 3-manifolds, which may aid in determining whether this construction can be done symplectically. This is a work in progress.

Krzysztof Święcicki
Title: Isomorphism problems between L^p and l^p spaces
Abstract: It is a natural question to ask, from the point of view nonlinear functional analysis, if L^p and l^p are coarsely equivalent. I will discuss the importance of this question to geometric group theory and mention some applications to geometric topology. Finally I'll go briefly over our result that there's no equivariant coarse embedding of L^p into l^p.

Linh Truong
Title: Truncated Heegaard Floer homology and concordance invariants
Abstract: I will discuss how to use a version of Heegaard Floer homology, called truncated HF, to construct a sequence of concordance invariants that generalize the Ozsvath-Szabo \nu invariant and the Hom-Wu \nu^+ invariant.

Michael Willis
Title: Stabilizations of link homologies for infinite braids
Abstract: The infinite full twist on n strands has been found to have many remarkable properties for many link homology theories. In this talk we will explain why, in all of these theories, any positive infinite braid (under very mild hypotheses) has the same stable link homology as the infinite full twist. In particular, this will mean that 'colored' versions of these link homology theories, which are often constructed using infinite twists, can be constructed just as well with any positive infinite braid. It also means that, for large positive braids, the corresponding link homology can be approximated using large torus braids, which have been computed in several cases and conjecturally hold even more interesting properties.

Rebecca Winarski
Title: The twisted rabbit problem and Hubbard trees
Abstract: The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one? After remaining open for 25 years, this problem was solved by Bartholdi---Nekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier and Margalit, we formulate the problem topologically and solve the problem using Hubbard trees.