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ISM discovery school 8–11 May 2018 + CIRGET conference 12–13 May 2018
Perspectives on bordered Heegaard Floer theory
Please note that this is a past event.
We are organizing a summer school focused on new developments in bordered Heegaard Floer theory and the consequences of these developments towards the so-called L-space conjecture. The school is paired with and will provide background for a 1½-day research conference.
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venue CIRGET, Montréal
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discovery school lecturers
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Jake Rasmussen, Cambridge, UK
CRM Simons Professor April–May 2018
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John Baldwin, Boston College, US
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Jonathan Hanselman, Princeton, US
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discovery school speakers
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Cameron Gordon, University of Texas at Austin, US
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Robert Lipshitz, Oregon, US
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Sarah Rasmussen, Cambridge, UK
CRM Simons Fellow April–May 2018
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conference speakers
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Xinghua Gao, University of Illinois, US
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Matt Hedden, Michigan State University, US
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Tom Hockenhull, Imperial College, UK
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Ying Hu, UQAM, CA
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Artem Kotelskiy, Princeton, US
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Marco Marengon, UCLA, US
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Ina Petkova, Dartmouth College, US
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Biji Wong, UQAM, CA
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organizers
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contact email address loopyloops2018\(@\)gmail.com
French translation missing!
All talks will be in SH-3620, which is a lecture theater in the Sherbrooke building. The Q&A exercise sessions will be in PK-5675, which is the new CIRGET room on the fifth floor of the President Kennedy building. Please see the red markers on the map.
A printer friendly version of the schedule can be found here.
A copy of the exercise sheet is available here.
| Please note that the order of the talks is still subject to change. |
| Tuesday |
| 10.00
| Jake Rasmussen |
lecture 1: Introduction to \(\operatorname{\widehat{HF}}\)
This lecture will be a basic introduction to Heegaard Floer homology. I'll
discuss \(\operatorname{\widehat{HF}}\) as a 3+1 dimensional TQFT, give its definition, and
outline some of its basic properties, including \(\operatorname{Spin^c}\) decomposition, the
exact triangle, and bounds on the genus of embedded surfaces.
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| 11.30
| John Baldwin |
lecture 1: 3-manifolds and open books
I'll introduce and motivate the notion of an open book as a useful tool for studying questions about 3-manifolds, and will explain connections between open books, braids, and the mapping class group of a surface.
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| 14.30
| John Baldwin |
lecture 2: Open books and the contact invariant
I'll introduce contact geometry and explain Giroux's correspondence between contact structures on 3-manifolds and open books. I'll then describe an invariant of contact structures, defined in terms of open books, which takes values in Heegaard Floer homology. This contact invariant and its properties can be used to study relationships between topological properties of open books and geometric properties of their corresponding contact structures. It can also be used to study the so-called L-space conjecture relating Heegaard Floer homology, taut foliations, and left-orderability.
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| 16.30
| Q&A exercise session @PK-5675
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| Wednesday |
| 9.30
| Jake Rasmussen |
lecture 2: \(\operatorname{\widehat{HF}}\) as a 2+1+1 dimensional TQFT
In this lecture, I'll consider \(\operatorname{\widehat{HF}}\) in the context of 2+1+1 dimensional
TQFT's. I'll explain the definition of cobordism map from this
perspective, and discuss bordered Floer homology, the Fukaya category of
\(\operatorname{Sym}^g(\Sigma\smallsetminus z)\), and the relation between them.
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| 11.30
| Jonathan Hanselman |
lecture 1: Bordered Floer homology for torus boundary
We will give a brief introduction to bordered Heegaard Floer homology as defined by Lipshitz, Ozsváth, and Thurston. To simplify the story, we will only consider the special case of manifolds whose boundary is a torus. We will define the bordered Heegaard diagrams that are used to encode such manifolds and discuss some of the algebra required for the definition. We will introduce (without giving all details of the definition) the type D structures and type A structures associated to a bordered manifold, and an algebraic pairing theorem.
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| 14.30
| John Baldwin |
lecture 3: Applications of the contact invariant
I'll discuss another, more Heegaard-diagrammatic description of the contact invariant, and describe some applications of open books and the contact invariant to problems involving Dehn surgery.
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| 16.30
| Q&A exercise session @PK-5675
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| Thursday |
| 9.30
| Jake Rasmussen |
lecture 3: Sutured Floer homology
I'll begin this lecture by discussing sutured manifolds and surface
decompositions. Next, I'll give several definitions of sutured Floer
homology and explain its relation to bordered Floer homology. Finally,
I'll discuss Juhász's surface decomposition theorem and describe some of
its applications.
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| 11.30
| Jonathan Hanselman |
lecture 2: Bordered Floer as immersed curves
We will discuss a geometric interpretation of the algebraic invariants introduced in the previous lecture. The modules CFD and CFA can be represented by a train track immersed in the boundary of the 3-manifold. Using a sequence of simplifications, which preserve the underlying bordered Floer information, these train tracks can be reduced to a collection of immersed curves in the boundary torus.
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| 14.30
| Jonathan Hanselman |
lecture 3: Properties and Applications of immersed curves
We will discuss a few properties of the immersed curves associated to a manifold with torus boundary, and see several examples. We will also see how the geometric approach to bordered invariants leads to easy proofs of some nice applications. For example, we will prove a lower bound on the size of \(\widehat{HF}\) for a 3-manifold containing an essential torus, and we will prove a gluing condition determining when gluing two manifolds along their torus boundaries produces an L-space.
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| 16.30
| Q&A exercise session @PK-5675
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| Friday |
| 9.30
| Jake Rasmussen |
lecture 4: Applications
I'll discuss applications of Floer homology for manifolds with torus
boundary to a topic related to the L-space conjecture. The exact topic
still to be decided. One possibility is to discuss the exact triangle,
nonvanishing of the Ozsváth-Szabó invariant for symplectic manifolds and
the fact that L-spaces do not admit taut foliations. Another is Floer
simple manifolds and the set of L-space Dehn filling slopes.
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| 11.30
| Cameron Gordon |
The L-space conjecture and cyclic branched covers of knots
The L-space conjecture asserts that for a closed, prime 3-manifold \(M\), the following are equivalent:
- \(\pi_1(M)\) is left-orderable,
- \(M\) supports a co-orientable taut foliation, and
- \(M\) is not a Heegaard Floer L-space.
We will discuss these properties in the case that \(M\) is of the form \(\Sigma_n(K)\), the \(n\)-fold cyclic branched cover of a knot \(K\). As well as summarizing known results, we will also describe several open problems.
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| 14.00
| Sarah Rasmussen |
title not available
abstract not available
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| 15.30
| Robert Lipshitz |
Bordered-sutured Floer homology and boundary-parallel tangles
We will sketch the construction of bordered-sutured Floer homology, focusing on the case of tangles. We will then discuss how bordered-sutured Floer homology makes (the "hat" variant of) knot Floer homology and, more generally, sutured Floer homology algorithmically computable and how this can be used to detect boundary-parallel tangles. This is joint work with Alishahi, building on earlier work of Zarev and joint work with Ozsváth-Thurston.
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| Saturday |
| 9.30
| Artem Kotelskiy |
Khovanov homology via immersed curves in the pillowcase
We will describe a geometric interpretation of Khovanov homology as Lagrangian Floer homology of two immersed curves in the pillowcase. For 2-bridge knots we will prove that the curves are natural in the sense that they are representation varieties of rational tangles. This is work in progress, and the building blocks for us will be:
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Pillowcase homology, which is a geometric construction developed by
[Hedden, Herald, Kirk] in order to better understand and compute a knot
invariant called singular instanton knot homology.
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A follow up construction by [Hedden, Herald, Hogancamp, Kirk], which
embeds bordered Khovanov theory into a certain subcategory of the Fukaya
category of the pillowcase.
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The structure result for the wrapped Fukaya category of a surface,
proved by [Haiden, Katzarkov, Kontsevich] and [Hanselman, Rasmussen,
Watson].
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| 10.50
| Ina Petkova |
Knot Floer homology and the \(\mathfrak{gl}(1|1)\) link invariant
The Reshetikhin-Turaev construction for the standard representation of the quantum group \(\mathfrak{gl}(1|1)\) sends tangles to \(\mathbb{C}(q)\)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology — a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.
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| 11.40
| Tom Hockenhull |
Heegaard Floer invariants of links
There are many ways to use the constructions which come under the umbrella of "Heegaard Floer homology" to associate an invariant to a link in the 3-sphere. In this talk, I will try to recount how some of these invariants are defined and interrelated, with an emphasis upon how the newest of these — the bordered-sutured Heegaard Floer homology of the complement of the link — is related to its older siblings.
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| 14.00
| Marco Marengon |
Strand algebras and bordered-style knot homologies
Inspired by bordered Floer homology, Ozsváth and Szabó introduced in 2016 a
knot invariant, which they announced to be isomorphic to the usual knot
Floer homology. The invariant is defined by tensoring bimodules over quiver
algebras. I will show that the quiver algebra is quasi-isomorphic to a
strand algebra on a chord diagram, and explain how we hope to build a
bridge between Ozsváth-Szabó's theory and bordered sutured Floer homology.
This is a joint work (in progress) with Mike Willis and Andy Manion.
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| 15.20
| Matt Hedden |
Irreducible homology \(S^1\times S^2\)s which aren't zero surgery on knot
I'll discuss constructions of manifolds with the homology of \(S^1\times S^2\) which don't arise as Dehn surgery on knots in \(S^3\). One of our obstructions comes from the Heegaard Floer d-invariants and has been around for a while but hadn't been successfully employed. Our examples have weight one fundamental group and were constructed to answer a question of Aschenbrenner, Friedl and Wilton. Moreover, they are not even homology cobordant to surgery on a knot in \(S^3\). This is joint work with Tom Mark, Kyungbae Park, and Min Hoon Kim.
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| Sunday |
| 9.30
| Ying Hu |
From taut foliations to the left-orderability of 3-manifold groups
The L-space conjecture asserts that for an irreducible, orientable three-manifold M, the following three statements are equivalent:
- M has a left-orderable fundamental group,
- M admits a co-orientable taut foliation and
- M is not a Heegaard Floer L-space.
This conjecture ties together the algebraic, topological and analytic aspects of three-manifold theory and also promotes new research directions.
In this talk, we will discuss progress made in the understanding of the implication from statement (b) to (a) above. We will show that the existence of certain "nice" taut foliations on a three-manifold leads to the left-orderability of its fundamental group. It is not clear whether three-manifolds that admit co-orientable taut foliations will admit one with such a "nice" property. However, large families of three-manifolds which do possess such "nice" taut foliations will be presented. As a result, the L-space conjecture is confirmed for these classes of three-manifolds. This is joint work with Steve Boyer.
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| 10.50
| Xinghua Gao |
Orderability of Dehn fillings
Boyer, Gordon, and Watson conjectured that an irreducible rational homology 3-sphere is not an L-space if and only if its fundamental group is left-orderable. In a recent work, Culler and Dunfield showed how to use \(\widetilde{PSL_2(\mathbb{R})}\) representations and the so called translation extension locus to construct orders on intervals of Dehn filling of certain homological solid torus. In this talk, I'll show how to use a similar technique to construct orders on Dehn fillings of some other homological solid torus.
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| 11.40
| Biji Wong |
A Floer homology invariant for 3-orbifolds via bordered Floer
Using bordered Floer, we construct an invariant for 3-orbifolds
\(Y^\text{orb}\) with singular set a knot that generalizes \(\widehat{HF}\) for 3-manifolds. We
show that for a large class of 3-orbifolds the invariant behaves like
\(\widehat{HF}\) in that the invariant (together with a relative \(\mathbb{Z}_2\)-grading)
categorifies the order of \(H_1^\text{orb}(Y^\text{orb})\). When \(Y^\text{orb}\) arises as Dehn surgery
on an integer-framed knot in \(S^3\), we use Jen Hom's \(\{-1,0,1\}\)-valued epsilon
knot invariant to prove a rank inequality between the orbifold invariant
and \(\widehat{HF}\) of the 3-manifold underlying \(Y^\text{orb}\).
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Those participants for whom we provide accommodation will be housed close to the workshop venue, namely at the Résidences de l'UQAM Ouest. Senior speakers are accommodated in junior suites at the Trylon apartment hotel. See the green markers on the map.
We ask all other participants to organize accommodation for themselves. We can suggest the following places:
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Trylon which is an apartment-hotel.
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Auberge du Jardin d'Antoine which is a small inn (breakfast is included).
At least one of the organizers can also report on excellent experiences with airbnb in Montréal.
travel by train.
UQAM is in walking distance from the central train station, where amtrak and via rail services connect Montréal to New York and major Canadian towns. See the yellow marker on the map.
travel by bus.
UQAM is in walking distance from the central bus station, where greyhound, limocar and others oligopolists meet. See the northernmost blue marker on the map.
travel by air.
The most convenient airport for the workshop is Montréal-P.-E.-Trudeau (YUL). At the arrivals, you can buy a bus ticket for $10 which allows unlimited travel throughout the STM bus and métro network and is valid for 24 consecutive hours. There are also other types of tickets available, see here. Outside the arrivals, follow the signs to the bus stop of the bus 747, which leaves regularly for the town center. For optimists among you, here is a schedule. The bus ride takes about 45-70 minutes. You can either get off at the central bus station, which is also the terminal stop, so that is easy. Alternatively, you can already get off at the bus stop Jeanne-Mance. In either case, it is only a short walk to UQAM. See also the blue markers on the map.
travel by car.
There's parking under the Résidences de l'UQAM Ouest. The entry point is on rue Saint-Urbain. However, owing to its very central location and proximity to Place des Arts, it is rather expensive (on average $24 per day, starting at $4 per 20 minutes).
Registration is now closed, since the event is over. Thanks everyone for coming!
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