# 41st Spring Lecture Series

"Submanifolds in Contact Topology." A series of five lectures by John Etnyre (Georgia Institute of Technology).

April 14-17, 2016

### Participants

John Baldwin (Boston College)
Roger Casals (Massachusetts Institute of Technology)
Chris Cornwell (CIRGET, Universite du Quebec a Montreal)
Emmy Murphy (Massachusetts Institute of Technology)
Olga Plamenevskaya (Stony Brook University)
Laura Starkston (Stanford University)
Bulent Tosun (University of Virginia)
Shea Vela-Vick (Louisiana State University)
Chris Wendl (University College London)

### 2016 Organizers

Matt Clay (University of Arkansas)
Yo'av Rieck (University of Arkansas)
Jeremy Van Horn-Morris (University of Arkansas)

The principal speaker for the Lecture Series is Professor John Etnyre, Georgia Institute of Technology, whose primary interest is the topology of low dimensional manifolds, its relationship with geometry and dynamics. Through a series of five talks, “Submanifolds in Contact Topology,” he will discuss how tools from low dimensional topology helped develop mathematicians’ understanding of contact manifolds and submanifolds in dimension three, and how those tools are being developed in higher dimensions.

The University of Arkansas Spring Lecture series are conferences organized every spring by the Department of Mathematical Sciences of the University of Arkansas. Each conference is focused on a specific topic chosen among the current leading research areas in Mathematics; a principal lecturer delivers a short, five-lecture course and selects a number of specialists who are invited to give talks on subjects closely related to the topic of the conference. Short talks by young Ph.D.s and finishing graduate students are solicited to complement the conference. Each Lecture Series has grown into an ideal opportunity for specialists and young researchers to meet and exchange ideas about topics at the forefront of modern mathematics.

The Spring Lectures are usually sponsored by the NSF jointly with the University of Arkansas. The proceedings of several conferences have appeared in the series "University of Arkansas Lecture Notes in the Mathematical Sciences," published by John Wiley & Sons.

John Etnyre (Georgia Institute of Technology)
Title: Transverse knots and the birth of 3 dimensional contact topology (Lecture 1)
Abstract: This series of talks will be about studying submanifolds of contact manifolds and using them to get a better understanding of contact geometry/topology. Before embarking on the general story, in this talk we will focus on the history of contact geometry in dimension three and the pivotal role transverse knots played in the development. This will allow us to introduce ideas in contact geometry for those who have had little experience with them and also indicate the type of problems in which one might be interested when trying to generalized the story to all dimensions.

John Etnyre (Georgia Institute of Technology)
Title: Embeddings of contact manifolds I  (Lecture 2)
Abstract: In this talk we will consider many general questions about contact embeddings (that is a generalization of transverse knots into higher dimensions). Many of these questions remain open, but in this talk we will discuss obstructions to finding contact embeddings and one way to actually find such embeddings. In particular we consider spun embeddings which generalize the idea of spun knots. Using this technique we will see that there are “nice” contact 5 manifolds into which all contact 3 manifolds embed.
This is joint work with Yanki Lekili.

John Etnyre (Georgia Institute of Technology)
Title: Embeddings of contact manifolds II (Lecture 3)
Abstract: In this talk we will consider many general questions about contact embeddings (that is a generalization of transverse knots into higher dimensions). Many of these questions remain open, but in this talk we will discuss obstructions to finding contact embeddings and one way to actually find such embeddings. In particular we consider spun embeddings which generalize the idea of spun knots. Using this technique we will see that there are “nice” contact 5 manifolds into which all contact 3 manifolds embed.
This is joint work with Yanki Lekili.

John Etnyre (Georgia Institute of Technology)
Title: Transverse surgery (Lecture 4)
Abstract: Transverse surgery was the original way in which contact structures were constructed on all 3-manifolds. In this talk we will discuss joint work with James Conway concerning the generalization of this idea to higher dimensions and then show that all (co-orinted) contact structures on all (oriented) 5-manifolds can be constructed through various surgery procedures (both transverse and Legendrian surgeries seem to be needed as of this moment).

John Etnyre (Georgia Institute of Technology)
Title: The contact isotopy problem (Lecture 5)
Abstract: In this talk we will consider various questions concerning the contact isotopy problem. This involves two basic questions (1) which smooth isotopy classes can be realized by a contact embedding and (2) when are to contact embeddings of one manifold into anther isotopic through contact embeddings. We will discuss a few obvious obstructions and attempts to create other obstructions, but will mainly discuss open problems in the area.

John Baldwin (Boston College)
Title: Stein fillings and SU(2) representations
Abstract: In recent work, Sivek and I defined invariants of contact 3-manifolds with boundary in sutured instanton Floer homology. I will sketch the proof of a theorem about these invariants which is analogous to a result of Plamenevskaya in Heegaard Floer homology: if a 4-manifold admits several Stein structures with distinct Chern classes, then the invariants of the induced contact structures on its boundary are linearly independent. As a corollary, we conclude that if a homology sphere Y admits a Stein filling with nonzero first Chern class, then there is a nontrivial representation from the fundamental group of Y to SU(2).
This is joint work in progress with Steven Sivek.

Roger Casals (MIT)
Title: Contact Topology from the Legendrian viewpoint I
Abstract: In this series of two talks we will discuss Legendrian fronts and their applications to contact and symplectic topology. These include the characterization of overtwisted contact structures in terms of Legendrian knots, the detection of flexible Weinstein structures and computation of wrapped Fukaya categories for affine Stein manifolds. This first talk is focused on the construction and proofs of the Front dictionary, translating exact Lagrangians expressed as Dehn twists to Legendrian links. In particular, the front presentation of a Dehn twist and Legendrian handle slides will be discussed, along with the basic features of higher-dimensional Legendrian calculus.
This is joint work with Emmy Murphy.

Chris Cornwell (L'Université du Québec à Montréal)
Title: Augmentation varieties
Abstract: I will discuss knot contact homology, a topological invariant defined through the canonical symplectic geometry of cotangent space, and the information we can extract from this theory through its augmentations. In particular, I will focus on a pair of varieties whose points come from an augmentation, and how these varieties relate to certain character varieties and to the A-polynomial.

Emmy Murphy (MIT)
Title: Contact Topology from the Legendrian viewpoint I
Abstract: In this series of two talks we will discuss Legendrian fronts and their applications to contact and symplectic topology. This second talk is focused on applications of the Front dictionary. These include the description of Stein handlebodies of affine algebraic varieties, the detection of flexible and subflexible Weinstein structures and the existence of exact Lagrangian submanifolds. In particular, we will compute Fukaya categories for a family of affine manifolds and deduce the mirror functor for the complement of a conic from a Legendrian perspective.
This is joint work with Roger Casals.

Olga Plamenevskaya (Stony Brook University)
Title: Transverse invariants and braid monodromy
Abstract: This talk will focus on transverse knots in the standard contact 3-space; up to stabilization, these knots can be represented by classical braids. Transverse knots have very subtle structure (beyond their topological type and the classical self-linking invariant); they can often be distinguished by an invariant in knot Floer homology, due to Ozsvath-Szabo-D.Thurston. We will show that non-vanishing of this Floer-homological invariant for a given transverse knot is related to the "amount of positive twisting" of the monodromy of its braid representative. The proof is an interplay of contact topology, combinatorics of braid words (via braid orderings theory), and grid diagrams.

Laura Starkston (Stanford University)
Title: Line arrangements in contact and symplectic topology
Abstract: A complex line arrangement is a collection of complex projective lines in CP2 which may intersect at points of multiplicity greater than two. We will discuss some similarities and differences between symplectic, topological, and complex algebraic line arrangements. These line arrangements give rise to certain contact manifolds and the symplectic realizability of the line arrangement determines the fillability of the contact manifold.
This is based on joint work with Danny Ruberman.

Bulent Tosun (University of Virginia)
Title: Contact Surgery and Naturality of Heegaard Floer Invariants.
Abstract: TBA

Shea Vela-Vick (Louisiana State University)
Title: A refinement of the contact invariant in Heegaard Floer theory
Abstract: We discuss refinements of Ozsvath and Szabo's contact invariant in Heegaard Floer theory. One such refinement, denoted b, takes values in the positive integers union infinity, and extends the usual contact invariant in the sense that if c(Y,\xi) is nonzero, then b is infinity. We further show that if (Y,\xi) is overtwisted, then b(Y,\xi) = 1, reflecting the usual vanishing of the usual contact invariant for such contact structures. We will focus our attention on the construction of b and discuss some of its basic properties and applications.
Everything discussed in this talk is joint work with John Baldwin.

Chris Wendl (University College London)
Title: Tight contact structures on connected sums need not be contact connected sums
Abstract: In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three. The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery.
This is joint work with Paolo Ghiggini and Klaus Niederkrüger.

Pan Yu (Duke University)
Title: The augmentation category map induced by exact Lagrangian cobordisms
Abstract: To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two ends. We study the functor and establish a long exact sequence relating the corresponding Legendrian cohomology categories of the two ends. As applications, we prove that the functor between augmentation categories is injective on objects, and find new obstructions to the existence of exact Lagrangian cobordisms.
The main technique is a recent work of Chantraine, Dimitroglou Rizell, Ghiggini and Golovko on Cthulhu homology.

SLS 2016 Video Archive Playlist

Reviel Netz (Stanford University)
"What Have We Learned from the Archimedes Palimpsest? (Public Lecture)"
SLS 2016 - 01

John Etnyre (Georgia Institute of Technology)
"Transverse knots and the birth of 3 dimensional contact topology (Lecture 1)"
SLS 2016 - 02

Chris Wendl (University College London)
"Tight contact structures on connected sums need not be contact connected sums"
SLS 2016 - 03

John Etnyre (Georgia Institute of Technology)
"Embeddings of contact manifolds I (Lecture 2)"
SLS 2016 - 04

Roger Casals (MIT)
"Contact topology from the legendarian viewpoint I"
SLS 2016 - 05

Roger Casals (MIT)
"Contact topology from the legendarian viewpoint II"
SLS 2016 - 06

John Etnyre (Georgia Institute of Technology)
"Embeddings of contact manifolds II (Lecture 3)"
SLS 2016 - 07

Shea Vela-Vick (Louisiana State University)
"A refinement of the contact invariant in Heegaard Floer theory"
SLS 2016 - 08

John Baldwin (Boston College)
"Stein fillings and SU(2) representations"
SLS 2016 - 09

John Etnyre (Georgia Institute of Technology)
"Transverse surgery (Lectures 4)"
SLS 2016 - 10

Bulent Tosun (University of Virginia)
"Contact Surgery and Naturality of Heegaard Floer Invariants"
SLS 2016 - 11

Ogla Plamenevskaya (Stony Brook University)
"Transverse invariants and braid monodromy"
SLS 2016 - 12

John Etnyre (Georgia Institute of Technology)
"The contact isotopy problem"
SLS 2016 - 13

Chris Cornwell (CIRGET, Universite du Quebec a Montreal)
"Augmentation varieties"
SLS 2016 - 14

Yu Pan (Duke University)
"The augmentation category map included by exact Lagrangian cobordisms"
SLS 2016 - 15

Laura Starkston (Stanford University)
"Line arrangements in contact and symplectic topology"
SLS 2016 - 16