PACT Seminar

Legendrian weaves, cluster mutation, and the botany problem

Legendrian weaves, cluster mutation, and the botany problem

Legendrian weaves, cluster mutation, and the botany problem

No PACT

PATCH at Swarthmore College

Science Center room 104*

*Note: This is a different room from last time!

Speakers: Anubhav Mukherjee (Princeton), Jean Pierre Mutanguha (Princeton & IAS)

10:00-11:00 am: An overview of mapping class groups of 4-manifolds, Anubhav Mukherjee.

11:30 am-12:30 pm: Free-by-cyclic groups from a tree's point of view, Jean Pierre Mutanguha.

12:30-2:30 pm: Lunch!

2:30-3:30 pm: An eye towards understanding of smooth mapping class groups of 4-manifolds, Anubhav Mukherjee. 

Abstract: The groundbreaking research by Freedman, Kreck, Perron, and Quinn provided valuable insights into the topological mapping class group of closed simply connected 4-manifolds. However, the development of gauge theory revealed the exotic nature of the smooth mapping class group of 4-manifolds in general. While gauge theory can at times obstruct smooth isotopy between two diffeomorphisms, it falls short of offering a comprehensive understanding of the existence of diffeomorphisms that are topologically isotopic but not smoothly so. In this talk, I will elucidate some fundamental principles and delve into the origins of such exotic diffeomorphisms. This is my upcoming work joint with Slava Krushkal, Mark Powell, and Terrin Warren.

4:00-5:00 pm Canonical hierarchical decompositions of free-by-cyclic groups, Jean Pierre Mutanguha

Abstract: Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss various dynamical properties of automorphisms that turn out to be group invariants of the corresponding free-by-cyclic groups (e.g. growth type). In particular, certain dynamical hierarchical decompositions of an automorphism determine canonical hierarchical decompositions of its mapping torus. In the intro talk, I will discuss how Bass-Serre theory (actions on simplicial trees) gives us a grip on these groups.

5:00-6:30 pm: Pizza dinner (probably, depending on interest)

Transportation

The best way to get to campus from Temple or Penn is probably to take the Media/ Wawa regional rail line to the Swarthmore stop, then walk through campus to the Science Center. But if you're driving, there is street parking on Elm Ave in between Walnut and N Chester, as well as some guest parking in Whittier Lot.

The best way to get to campus from Haverford or Bryn Mawr is probably to take the TriCo shuttle, which will drop you off in the center of campus. If you're driving and have a Bryn Mawr or Haverford faculty parking pass, you can park in any of the lots-- the science center adjacent parking lots are Whittier, Dupont, and Water Tower.

Contact surgery and symplectic fillability

Abstract: It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties (e.g. tightness, fillability, vanishing or non-vanishing of various Floer or symplectic homology classes) of contact structures are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. The case of positive contact surgeries much more subtle. In this talk, extending an earlier work of the speaker with J. Conway and J. Etnyre, I will discuss some new results, jointly with T. Mark,  about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (positive) integer surgery along a Legendrian knot to yield a fillable contact manifold. I will in particular focus on examples, applications and related open questions. 

Relative Calabi-Yau Structures for Legendrian Contact Homology

Abstract: Legendrian Contact Homology (LCH) was among the first, and is still among the most important, non-classical invariants of Legendrian knots.  In this talk, I will tell a story about how to understand ever more of the algebraic structure of LCH. I promise pictures and examples, and I hope to explain the words in the title by the end of the talk.  This is joint work with Jason Ma and, towards the end, work in progress in conversation with Zhenyi Chen.

Relative Calabi-Yau Structures for Legendrian Contact Homology (Part 2 of 1)

Contact surgery and symplectic fillability (Part 2)

The Geography of Immersed Lagrangian Fillings of Legendrian Submanifolds

Abstract: A classic problem in topology is to fix a smooth knot in the 3-sphere and search for smooth “filling” surfaces in the 4-ball that have that knot as its boundary. In the contact world, we add some geometric constraints: the knot is now Legendrian, and the filling is Lagrangian.  In fact, one can consider the problem in higher dimensions, where one searches for Lagrangian fillings of an n-dimensional Legendrian submanifold.  In the embedded setting, Sabloff and I proved a generating family version of what is commonly known as the Seidel isomorphism.  Informally, this isomorphism says that there are invariants of the Legendrian that can obstruct the existence of an embedded Lagrangian filling.  I will discuss work, joint with Samantha Pezzimenti, that show that these same Legendrian invariants that obstruct the existence of an embedded filling also give information about the combinations of Betti numbers and transverse double points that can be realized by immersed Lagrangian fillings. 

The Geography of Immersed Lagrangian Fillings of Legendrian Submanifolds, Part II

PATCH in the Tri-Co

Speakers: Michael Landry (Saint Louis University) & Hiro Lee Tanaka (Texas State University)

Morning Background Talks (Park 278)

• 10 – 11 “A stack of broken lines: Like BG, but for Morse Theory”, Tanaka
• Abstract: I will discuss some needed background for the afternoon talk.
• 11 – 11:30 Break
• 11:30 – 12:30 “Objects associated with 3-manifolds fibering over the circle”, Landry

Abstract: A fundamental example in low-dimensional topology is a closed oriented 3-manifold fibering over the circle. Thurston's study of this example led to the celebrated Nielsen-Thurston classification of surface homeomorphisms and the Thurston norm on homology. I will introduce these concepts before further developing some of the rich structure present in the example, touching on flows, foliations, and homeomorphisms of surfaces with infinitely generated fundamental group. I will mention joint work with Minsky and Taylor that fits into the story.

Afternoon Talks (Park 336)

• 2:30-3:30, “A stack of broken lines: Like BG, but for Morse Theory”, Tanaka

Abstract:  A G-bundle over X is a family of copies of G, with one copy for every element of X. Families like this arise when studying nice functions on manifolds (i.e., in Morse theory) -- where instead of families of groups, families of broken lines live on moduli of gradient trajectories. And just like G-bundles are classified by an object called BG, it turns out you can write down the object that classifies families of broken lines -- this object is the stack of broken lines. The amazing fact is that this (geometric) object has an incredibly deep connection to the (algebraic) idea of associativity, and I'll try to explain why this is true. If time allows (which it might not) I'll try to explain why this object is expected to play a central role in enriching Morse theory and various Floer theories over stable homotopy theory. This is joint work with Jacob Lurie.

• 3:30 – 4 Break
• 4-5 “Toward a dynamical theory of Thurston's norm”, Landry

Abstract: One might hope to generalize the picture described in the morning talk to the setting of 3-manifolds that do not necessarily fiber over the circle. I will give some of the history of this endeavor, mentioning three conjectures of Mosher from the 1990s. Then I will describe joint work with Tsang that aims to make progress on these conjectures using modern objects called veering branched surfaces.

The Geography of Immersed Lagrangian Fillings of Legendrian Submanifolds, Part III (Conclusion)

A Low-Tech Approach to Contact Nonsqueezing

Gromov's non-squeezing theorem can be paraphrased to say that in dimension 4 or higher, the standard ball cannot be symplectically embedded (squeezed) into a cylinder with a smaller radius. This symplectic non-squeezing theorem established symplectic rigidity and is widely regarded as one of the most important theorems in symplectic geometry. 

In contrast, in the contact setting in all dimensions, a standard ball of any radius can be contact embedded into an arbitrarily small neighborhood of a point. However, Eliashberg, Kim and Polterovich [EKP] discovered contact rigidity for pre-quantized balls in R2n x S1 under a more restrictive notion of squeezing - embeddings via a compactly supported ambient contactomorphisms.  In particular, in 2006 [EKP] showed that squeezing is possible at small scales even via contactomorphisms isotopic to the identity, while, at large scale, meaning 1 < R1 < R2, if there exists an integer between R1and R2, then there does not exist a contact squeezing of the pre-quantized ball of radius R2 into the pre-quantized ball of radius R1.  In 2017, Chiu applied the technique of microlocal sheaves to show that squeezing by a contact isotopy is impossible at large scale (1 < R1 < R2) regardless of whether there is an integer between R1 and R2.  In 2016, I proved large scale squeezing is impossible (even without the condition of isotopy to the identity); the argument used an equivariant version of the contact homology used by Eliashberg, Kim and Polterovich. Very recently, in joint work with Sandon and Zhang, we give an alternate proof of Chiu’s result by developing Zk-equivariant GF-homology for a synthetic Zk-action. In today’s talk, I will sketch another proof, one that again uses the theory of generating families, in fact only the classical GF homology groups defined by Sandon (2011) based on those of Traynor (1994). The argument uses the theory of continuation maps and persistence modules.  This is joint work with Lisa Traynor.

PATCH @ Temple

Speakers: Tam Cheetham-West, Yale University and Siddhi Krishna, Columbia University

• 9:30 am: Tam Cheetham-West background talk

"Using finite groups to approximate infinite groups"

Abstract: Some infinite groups like the group of integers with addition have lots of finite quotients. What can we use these finite quotients to do? What do these collections of finite quotients remember about the groups that produce them?

• 11:30am: Siddhi Krishna background talk

"Fibered knots: what, why and how"

Abstract: Fibered knots show up all over low-dimensional topology, as they provide a robust way to investigate interactions between phenomena of different dimensions. In this talk, I'll survey what they are, why you should care, and how to identify them. Then, as time permits, I'll also sketch a proof that positive braid knots are fibered. I will assume very little background for this talk -- all are welcome!

• 12:30pm: Lunch break
• 2:30pm: Tam Cheetham-West research talk

"Finite quotients of fibered, hyperbolic 3-manifold groups"

Abstract: The finite quotients of the fundamental group of a 3-manifold are the deck groups of its finite regular covers. We often pass to these finite-sheeted covers for different reasons, and these deck groups are organized into a topological group called the profinite completion of a 3-manifold group. In this talk, we will discuss how to leverage certain properties of mapping class groups of finite-type surfaces to study the profinite completions of the fundamental groups of fibered hyperbolic 3-manifolds of finite volume.

• 4:00pm: Siddhi Krishna research talk

"Braid positivity, taut foliations, and unknot detection"

Abstract: The L-space conjecture predicts that three seemingly different ways to measure the "size" of a 3-manifold are equivalent. In particular, it predicts that a manifold with the "extra" geometric structure of a taut foliation also has "extra" Heegaard Floer homology. In this talk, I'll discuss the motivation for this conjecture, and describe some new results which produce taut foliations by leveraging special properties of positive braid knots. Along the way, we will produce some novel obstructions to braid positivity. I will not assume any background knowledge in Floer or foliation theories; all are welcome!

A partial solution to Hedden's conjecture

Abstract: A pattern, or knot in a solid torus, induces a map on the set of knots modulo smooth concordance. In 2016, Matt Hedden conjectured that essentially none of these maps are group homomorphisms--more precisely, that if $P(K_1\# K_2)$ is always concordant to $P(K_1) \# P(K_2)$,  then the map induced by $P$ is either the identity map, the reversal map, or the zero map. In particular, this would imply that patterns with winding number not in the set {-1,0,1}  definitely do not induce homomorphisms. I will discuss work with Randall Johanningsmeier (Swat '24) and Hillary Kim (Swat '25) in which we prove that if $P$ is a pattern with winding number that is even but not divisible by eight, then P cannot induce a homomorphism on the smooth concordance group. This relies on previous work with Tye Lidman and Juanita Pinzon-Caicedo, and is the first result that obstructs all patterns of a given winding number from inducing homomorphisms.

A partial solution to Hedden's conjecture (continued and conclusion).

No meeting: Happy Thanksgiving!

Relations in Legendrian Knot Theory

Abstract:  Cobordisms between links, and the classical concordance group in particular, are a central framework in knot theory.  In this talk, I will compare and contrast structures arising from smooth cobordism between smooth knots to those that appear in the world of contact and symplectic topology, namely Lagrangian cobordism between Legendrian knots.  What type of relations are these?  Is there any algebraic structure?  What about metric structure? The bias will be towards open questions and sample results rather than a deep dive into any particular aspect of the theory. I will discuss joint work with Shea Vela-Vick, Mike Wong, and Angela Wu, as well as with Lisa Traynor.