# PACT Seminar

PACT (Philadelphia Area Contact Topology) Seminar

**Welcome to Math PACT Seminars**

The PACT seminar is a weekly gathering of faculty and students from the **P**hiladelphia **A**rea interested in (**C**ontact) **T**opology.

**Coffee and refreshments at 8:20-8:30 a.m. in Park 361. Talk at 8:30-9:45 a.m. in Park 336.**

**Spring 2022**

**Tuesday, Feb. 1**

**Speaker**: Allison Miller (Swarthmore)

**Title**: Slicing knots in definite 4-manifolds

**Abstract**: A knot is called "slice" if it bounds an embedded disc in the 4-ball. A natural extension of this idea is to think about knots that bound embedded discs in other simple 4-manifolds. We'll talk about some constructions and obstructions in the specific case of connected sums of the complex projective plane. Tools will involve a little bit of Kirby calculus, Donaldson's theorem about the intersection form of smooth definite 4-manifolds, and Freedman's result that knots with trivial Alexander polynomial are topologically slice.

**Tuesday, Feb. 8**

**Speaker**: Allison Miller (Swarthmore)

**Title**: Slicing knots in definite 4-manifolds (Part 2)

**Tuesday, Feb. 15**

**Speaker**: Allison Miller (Swarthmore)

**Title**: Slicing knots in definite 4-manifolds

**Tuesday, Feb. 22**

**Speaker**: Angela Wu (Louisiana State University)

**Title**: Obstructing Lagrangian concordance for closures of 3-braids

**Abstract**: Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself. The proof uses a variety of techniques in contact and symplectic geometry, from open book decompositions and Weinstein handle diagrams, to symplectic homology and the Legendrian contact homology DGA.

**Tuesday February 29**

**Speaker**: Angela Wu (Louisiana State University)

**Title**: Obstructing Lagrangian concordance for closures of 3-braids (continued)

**Tuesday March 15**

**Speaker**: Joshua Sabloff (Haverford College)

**Title**: On the Lagrangian cobordism relation on Legendrian links

**Abstract**: Lagrangian cobordism induces a preorder on the set of Legendrian links in any contact 3-manifold. We show that any finite collection of null-homologous Legendrian links in a tight contact 3-manifold with a common rotation number has an upper bound with respect to the pre-order. While a similar result due to Lazarev in higher dimensions requires the use of an h-principle, we are able to work "by hand" in dimension 3. In particular, we give concrete constructions of an exact Lagrangian cobordism from each element of the collection to a common Legendrian link. This construction allows us to define a notion of minimal Lagrangian genus between any two null-homologous Legendrian links with common rotation number, a notion that I will explore further towards the end of the talk. Most of this material is joint work with Shea Vela-Vick and C.-M. Michael Wong.

** **

**Tuesday March 22**, 8:30, In person or remote

**Speaker**: Josh Sabloff (Haverford College)

**Title:** Non-Orientable Lagrangian Fillings of Legendrian Knots

**Abstract:** After discussing some questions about Lagrangian Quasi-Cobordism left over from last week, we will discuss the following: Most work on Lagrangian fillings to date has concentrated on orientable fillings, but instead I will present some first steps in constructions of and obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings. This is joint work in progress with Linyi Chen, Grant Crider-Philips, Braeden Reinoso, and Natalie Yao.

**In Person:** Park 338

**Tuesday March 29**, 8:30, In person or remote

**Speaker**: Josh Sabloff

**Title:** Non-Orientable Lagrangian Fillings of Legendrian Knots (continued)

**Friday April 1, PATCH @ Temple**

9:30 and 2:30: Katie Mann

11:00 and 4:00: Inbar Klang

Titles and abstracts are below.**Klang Title: **Manifold topology via isovariant homotopy theory**Klang Abstract:**

Homotopy theory has proven to be a robust tool for studying non-homotopical questions about manifolds; for example, surgery theory addresses manifold classification questions using homotopy theory. In joint work with Sarah Yeakel, we are developing a program to study manifold topology via isovariant homotopy theory. I'll explain what isovariant homotopy theory is and how it relates to the study of manifolds via their configuration spaces, and talk about an application to fixed point theory.

In the morning background talk (at 11:00am), I will talk about configuration spaces and about homotopical fixed point theory. These are two relevant examples of how homotopy theory is used in manifold topology.

*****Mann Title: **Anosov flows, foliated planes, and ideal circles**Mann Abstract: **From an Anosov flow on a 3-manifold, one can extract an action of the fundamental group of the manifold on a plane preserving a pair of transverse foliations, and on a compactification of the plane by an ideal circle. My talks will give an introduction to this picture and show a recent application, joint with Thomas Barthelme and Steven Frankel on the classification problem for Anosov flows. By proving rigidity results about group actions on planes and circles, we show that transitive (pseudo-)Anosov flows are determined (up to orbit equivalence) by the algebraic data of the set of free homotopy classes of closed orbits.

In the morning background talk (at 9:30am), I will give an introduction to basic examples and structure theory of Anosov flows on 3-manifolds, focusing on the relationship between the geometry and topology of a manifold and the possible examples of flows it admits.

****

**Tuesday April 5 [No PACT Meeting]**

****

**Tuesday April 12 & Tuesday April 19**

**Speaker: **Hannah Schwartz (Princeton)

**Title: Doubles of Gluck twists**

**Abstract**: Among the most well-known potential counterexamples to the smooth, 4-dimensional Poincare conjecture is a family of homotopy 4-spheres called "Gluck twists", each of which is constructed by removing and regluing the regular neighborhood of a knotted 2-sphere smoothly embedded in the 4-sphere. called "Gluck twists". These manifolds were defined by Gluck in the 60's; since then, a great deal of effort has been spent understanding which knotted 2-spheres have Gluck twists diffeomorphic to the standard 4-sphere. Throughout the next two weeks, we will consider Gluck twists of spheres of the form K # - K for a 2-knot K and discuss joint work (in progress with Gabai and Naylor) that under certain assumptions on K these manifolds are standard. We will take plenty of time to develop the techniques used in our argument, made distinct from most previous strategies used to show that Gluck twists are standard by passing from a 4-dimensional to a 5-dimensional setting.

PATCH @ Penn, sponsored jointly with Bryn Mawr, Haverford and Temple.

**Friday April 22**

**9:30 AM** in DRL A2 Hannah Schwartz "Pretty pictures of regular homotopies"

This introductory talk will be largely picture-based, and will offer geometric interpretations of

the Freedman-Quinn and Dax invariants relevant to the afternoon lecture. In particular, we will

showcase examples in which each does not vanish and so obstructs isotopy between pairs of

homotopic surfaces.

**11 AM** in DRL A2 Franco Vargas Pallete "Volume in Hyperbolic Geometry"

We will see the role that volume plays in hyperbolic geometry. We will prove a version of

volume rigidity and introduce the Bonahon-Schlafli formula. Both statements will prove

extremely useful for the second part of this talk.

**12 - 2 PM** in 4E17 Provide lunch here and go outside to eat

**2 PM **in DRL A2 Hannah Schwartz "Isotopy vs. homotopy for disks with a common dual"

Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic

surfaces with a common dual has reinvigorated the study of concordance invariants defined by

Freedman and Quinn in the 90's, along with homotopy theoretic isotopy invariants of Dax from

the 70's. We will outline, give context to, and discuss techniques used to prove these so called

"light bulb theorems", and present new light bulb theorems for disks rather than spheres.

**3:30 PM** in DRL A2 Franco Vargas Pallete

"Peripheral birationality for 3-dimensional convex co-compact PSL(2, C) varieties"

It is a consequence of a well-known result of Ahlfors and Bers that the PSL(2, C) character

associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data.

In this talk we will show how this map extends to a birational isomorphism of the corresponding

PSL(2, C) character varieties, so in particular it is generically a 1-to-1 map. Analogous results

were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume

hyperbolic 3-manifolds. This is joint work with Ian Agol.

The speakers are Hannah Schwartz from Princeton and Franco Vargas Pallete from Yale.

All talks will also be streamed live and recorded on Zoom:

https://upenn.zoom.us/j/97944630643

**Fall 2021**

**Tuesday September 14, 2021**

Speaker: Isaac Sundberg (Bryn Mawr)

Title: The Khovanov homology of slice disks

Abstract: A smooth, oriented surface that is properly embedded in the 4-ball can be regarded as a cobordism between the links it bounds, namely, the empty link and its boundary in the 3-sphere. To such a link cobordism, there is an associated linear map between the Khovanov homology groups of its boundary links, and moreover, this map is invariant up to boundary-preserving isotopy of the cobordism. In this sequence of talks, we discuss these maps and use their invariance to understand the existence and uniqueness of slice disks and other surfaces in the 4-ball. This reflects joint work with Jonah Swann and with Kyle Hayden.

**Tuesday September 21, 2021**

Speaker: Isaac Sundberg (Bryn Mawr)

Title: The Khovanov homology of slice disks (Part II)

** ****Tuesday September 28, 2021**

Speaker: Isaac Sundberg (Bryn Mawr)

Title: The Khovanov homology of slice disks (Part III)

**Friday October 1, 2021**

**PATCH @ Bryn Mawr College, Park Science Center, Room 159**

The PATCH Seminar is a monthly gathering of faculty and students in the **P**hiladelphia **A**rea interested in **T**opology, **C**ontact/Symplectic Topology, and **H**yperbolic Geometry. This is an all-day event featuring two guest speakers.

**Speakers: Emmy Murphy (Princeton & IAS), Emily Stark (Wesleyan)**

**Intro/Background Talk #1**: **10 -11, Emily Stark**

**Title**: Geometry of finitely generated groups

**Abstract**: In the 1980s Gromov proposed studying finitely generated groups as metric spaces. This perspective is powerful as groups that have similar large-scale geometry often share common algebraic features. In this introductory talk, we will present examples of this phenomena as well as tools to study the geometry of a finitely generated group.

**Intro/Background Talk #2**: **11:30 – 12:30, Emmy Murphy**

**Title: **Weinstein handles and flexibility

**Abstract: **We'll discuss the basics of Liouville manifolds and Weinstein handles. This is a method by which new symplectic manifolds can be constructed from old, using isotropic/Legendrian submanifolds of contact manifolds. We'll also discuss some of the ways this interacts with contact flexibility, namely loose Legendrians and overtwisted contact structures. These are tools by which, using some semi-local hypotheses, the geometric structures in question can be completely understood in terms of smooth topology.

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**Lunch**:** 12:30 – 3**.

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**Research Talk #1: 3-4, Emily Stark**

**Title:** Graphically discrete groups and rigidity

**Abstract:** Rigidity theorems prove that a group's geometry determines its algebra, typically up to virtual isomorphism. Motivated by interest in rigidity, we study the family of graphically discrete groups. In this talk, we will present rigidity consequences for groups in this family. We will present classic examples as well as new results that imply this property is not a quasi-isometry invariant. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

**Research Talk #2: 4:30 – 5:30, Emmy Murphy**

**Title: **Liouville cobordisms

**Abstract: **In this talk we'll discuss some interesting Liouville cobordisms arising in the particular case when the negative boundary is an overtwisted contact manifold. This will center on two independent constructions: concordances in the high-dimensional setting, and cobordisms with high-index (and therefore non-Weinstein) topological type.

**Tuesday October 5, 2021 [Seminar meets in Park 337 today]**

Speaker: Isaac Sundberg (Bryn Mawr)

Title: The Khovanov homology of slice disks (Part IV)

**Tuesday October 12, 2021: No Meeting [Fall Break]**

**Tuesday October 19, 2021 **

Speaker: Isaac Sundberg (Bryn Mawr)

Title: The Khovanov homology of slice disks (Part V): Conclusion

PATCH @ Penn

Thursday 10/21

**Tuesday October 26, 2021: No meeting this week**

**Tuesday November 2, 2021:**

Speaker: Ziva Myer

Title: Constructing Lagrangian Cobordisms

Abstract: Just as smooth cobordisms help us understand and relate knots, in the more geometric setting of contact/symplectic geometry, Lagrangian cobordisms impose an interesting relation on Legendrian knots. Most constructions of Lagrangian cobordisms use "elementary" building blocks that are specific 0-handles and 1-handles. An important question is if every pair of knots that are connected by an exact Lagrangian cobordism can also be connected by one that is ribbon, or more specifically "decomposable" into these elementary moves. I will outline a strategy to answer this question that involves constructing a cobordism that is obstructed from being decomposable. This work is joint with Sarah Blackwell, Noémie Legout , Caitlin Leverson, Maÿlis Limouzineau, Yu Pan, Samantha Pezzimenti, Lara Simone Suárez, and Lisa Traynor.

The PACT Seminar is funded by the Mellon Tri-College Faculty Forum Program and hosts research talks on a broad range of topics from contact/symplectic topology, low-dimensional topology, and algebraic topology. The seminar is jointly organized by Professor Thomas Hunter (Swarthmore), Professor Paul Melvin (Bryn Mawr), Professor Josh Sabloff (Haverford), and Professor Lisa Traynor (Bryn Mawr). We are a friendly and mathematically engaged group composed of members of mathematics departments around the Philadelphia area including Bryn Mawr, Haverford, Swarthmore, Penn, Penn State Brandywine, Widener, and Temple.

**Goals for the seminar are:**

- To explore new developments in contact, symplectic, and low-dimensional topology;
- To encourage collaboration between participants, and to find entry points for such collaboration through the exchange of ideas and works in progress;
- To involve advanced undergraduate and graduate students in ongoing research in a seminar format; and
- To share, support, and critique work in progress.

The PACT seminar typically meets weekly (Tuesday mornings!) to enable detailed discussions of the participants’ own work or topics of particular interest to the participants. The seminar is usually conducted in a “mini-course” format, in which one member typically spends three to six sessions explaining research ideas in more depth than is feasible during a one-time seminar, and with more of an emphasis on underlying ideas than can easily be gleaned from a formal paper.

The PACT seminar meets on Bryn Mawr's beautiful campus, which is easy to reach from any of Philadelphia's campuses by car or train. The talk is from 8:30-9:45 a.m. in Park 336, and is preceded by coffee and breakfast refreshments at 8:15 a.m. in the math lounge, Park 361.

## Department of Mathematics

Contact Us

Park Science Building

Bryn Mawr College

Bryn Mawr, Pennsylvania 19010-2899

Phone: 610-526-5348

Fax: 610-526-6575

Tina Fasbinder

Academic Administrative Assistant

tfasbinder@brynmawr.edu

610-526-5348