



3:304:00 

Tea 


4:004:50 

Colloquium: Emmy Murphy 

Planar graphs and Legendrian surfaces Scanned Notes

5:055:50 

Lightning talk session 1 

4 talks see below
Slides

6:00 

Social Gathering 

Club Room at the Georgia Tech Hotel





9:009:30 

Light refreshments 


9:3010:20 

Talk: Tye Lidman 

Band surgeries on the trefoil Scanned Notes 
10:2011:00 

Tea Break 


11:0011:50 

Talk: Tori Akin *** 

Automorphisms of the Punctured Mapping Class Group 
11:0011:50 

Talk: Lei Chen 

The number of fiberings of a surface bundle over a surface Scanned Notes 
11:5012:30 

Catered Lunch 

Fox Bros BBQ 
12:301:30 

Lightning talk session 2 

8 talks see below
Slides

1:302:00 

Break 


2:002:50 

Talk: Balázs Strenner 

Fibrations of 3manifolds, tilings and nowhere continuous functions Scanned Notes 
2:504:10 

Extended Tea Break 


4:105:00 

Talk: Diana Hubbard 

The braid index, the fractional Dehn twist coefficient, and Upsilon Scanned Notes 
6:00 

Banquet 

At: South City Kitchen





9:009:30 

Light refreshments 


9:3010:20 

Talk: Justin Lanier 

Three theorems about generating mapping class groups Slides from talk 
10:2511:10 

Lightning talk session 3 

7 talks see below
Slides

11:1011:40 

Tea Break 


11:4012:30 

Talk: Ciprian Manolescu 

A sheaftheoretic model for SL(2,C) Floer homology Scanned Notes 
All talks are in Skiles 006.
*** Cancelled due to weather.

Michael Dougherty 
Braids with BoundaryParallel Strands 

Michael Willis 
Stabilizations of link homologies for infinite braids 

Bakul Sathaye 
Link obstruction to Riemannian smoothings of a locally CAT(0) manifold 

Jonathan Paprocki 
Topological Quantum Compiling via Quantum Teichmuller Theory 

Krzysztof Swiecicki *** 
Isomorphism problems between L^p and l^p spaces 

Bahar Acu 
Fillings of iterated planar contact manifolds 

Sudipta Kolay 
Generalized Alexander's Theorem 

Ryan Leigon 
An Excursion in Gluing Maps 

Aamir Rasheed 
Essential embeddings and immersions of surfaces in a 3manifold 

Haofei Fan 
Unoriented cobordism maps on link Floer homology 

Becca Winarski 
The twisted rabbit problem and Hubbard trees 

Jonathan Simone 
Towards a new construction of exotic 4manifolds 

Kevin Kordek 
A large abelian quotient of the level 4 braid group 

Sarah Davis 
Covering spaces, mapping class groups, and the symplectic representation 

Joshua Pankau 
Salem number stretch factors 

Adam Saltz 
Why are there so many spectral sequences from Khovanov homology? 

Marissa Loving 
Least dilatation of pure surface braids 

Linh Truong 
Truncated Heegaard Floer homology and concordance invariants 

Yu Pan 
Augmentations and immersed exact Lagrangian fillings 

Peter LambertCole 
Trisections of complex surfaces 
Titles and Abstracts
Automorphisms of the Punctured Mapping Class Group
We can describe the pointpushing 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 IvanovMcCarthy establishing the triviality of Out(Mod±). To this end, we’ll demonstrate that the pointpushing subgroup is “unique” in the mapping class group.
The number of fiberings of a surface bundle over a surface
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 4manifold M to have SFib(M)=2. I will then apply the criterion to classify the number of fiberings of the famous AtiyahKodaira manifold and any finite cover of a trivial surface bundle.
The braid index, the fractional Dehn twist coefficient, and Upsilon
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 nbraid has fractional Dehn twist coefficient greater than n1 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.
Three theorems about generating mapping class groups
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 pseudoAnosov mapping classes with sufficiently small stretch factor. Our pseudoAnosov examples answer a question of Darren Long from 1986.
Band surgeries on the trefoil
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.
A sheaftheoretic model for SL(2,C) Floer homology
I will explain the construction of a new homology theory for threemanifolds, 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 KapustinWitten equations and Khovanov homology. This is joint work with Mohammed Abouzaid.
Planar graphs and Legendrian surfaces
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 pseudoholomorphic 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 ncolorings which are strongly nonlinear 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.
Fibrations of 3manifolds, tilings and nowhere continuous functions
We start with a 3manifold fibering over the circle and investigate how the pseudoAnosov 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.
Lightning talks
Fillings of iterated planar contact manifolds
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 3dimensional results regarding symplectic fillings (cobordism from the empty set to a contact manifold) of planar contact manifolds and speculate some higherdimensional preliminary results. This is a joint work in progress with J. Etnyre and B. Ozbagci.
Covering spaces, mapping class groups, and the symplectic representation
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 3fold 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.
Braids with BoundaryParallel Strands
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.
Unoriented cobordism maps on link Floer homology
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}thStipsiczSzab\'{o}’s and Manolescu’s constructions of link cobordism maps for an unoriented band move.
Generalized Alexander's Theorem
A classical theorem of Alexander about closed braids allows us to study knots in threespace using the theory of braids. In this talk, we will define closed braids in higher dimensions, and discuss generalizations of Alexander's theorem.
A large abelian quotient of the level 4 braid group
It is generally a difficult problem to compute the Betti numbers of a given finiteindex 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.
An Excursion in Gluing Maps
Sutured Floer homology is a powerful 3manifold 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, HondaKazezMatic 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.
Least dilatation of pure surface braids
The nstranded 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 npunctures 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 pseudoAnosovs due to Dowdall and Aougab—Taylor. I am interested in the least dilatation of pseudoAnosov 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.
Augmentations and immersed exact Lagrangian fillings
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.
Salem number stretch factors
Associated to every pseudoAnosov homeomorphism of a closed orientable surface is a real number called the stretch factor. Thurston showed that the stretch factor of any pseudoAnosov 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 pseudoAnosov maps due to Thurston, and a particular algebraic unit known as a Salem number.
Topological Quantum Compiling via Quantum Teichmuller Theory
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) ChernSimons theory as a special case. The solution to this problem has applications in topological quantum computing.
Essential embeddings and immersions of surfaces in a 3manifold
In this talk we will discuss a theorem which gives a sufficient condition for two embedded surfaces in an irreducible 3manifold to be isotopic. We shall also discuss some generalizations, such as how to detect whether two essential immersions of surfaces are homotopic.
Why are there so many spectral sequences from Khovanov homology?
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 KhovanovFloer theories and prove some cool facts about them using spectral sequences. I'll discuss techniques to prove stronger theorems by avoiding spectral sequences altogether.
Link obstruction to Riemannian smoothings of a locally CAT(0) manifold.
By work of Davis, Januszkiewicz and Lafont, there exist obstructions to having a smooth Riemannian metric of nonpositive 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 2dimensional flats with the property that their boundaries form a nontrivial link.
Towards a new construction of exotic 4manifolds
In this talk I will introduce a new construction (similar to the rational blowdown) that can potentially be used build exotic 4manifolds. In particular, I will describe the machinery I have developed and potential examples of new exotic 4manifolds. I will also describe a recent result that classifies tight contact structures with no Giroux torsion on plumbed 3manifolds, which may aid in determining whether this construction can be done symplectically. This is a work in progress.
Isomorphism problems between L^p and l^p spaces
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.
Truncated Heegaard Floer homology and concordance invariants
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 OzsvathSzabo \nu invariant and the HomWu \nu^+ invariant.
Stabilizations of link homologies for infinite braids
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.
The twisted rabbit problem and Hubbard trees
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 BartholdiNekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier and Margalit, we formulate the problem topologically and solve the problem using Hubbard trees.
