Columbia Geometric Topology Seminar

Fall 2023


Organizers: Ross Akhmechet, Siddhi KrishnaFrancesco Lin

The GT seminar typically meets on Fridays at 2:00pm Eastern time in Room 307, Mathematics Department, Columbia University. 

On Friday, October 13th, the seminar will be streamed over Zoom. Here is the Zoom link. 

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Spring 2023

Date Time (Eastern) Speaker Title

September 8

2pm Eastern

Liam Watson

Khovanov multicurves are linear

September 15

2pm Eastern

Anna Parlak

Faces of the Thurston norm ball dynamically represented by multiple distinct flows

September 22

2pm Eastern Matt Hogancamp

Khovanov homology and handleslides

September 29

2pm Eastern


no seminar today!

October 6

2pm Eastern

Inanc Baykur

Exotic 4-manifolds with signature zero

October 13

4:30pm Eastern in Room 312

Ian Agol

Ribbon concordance of knots is a partial order

October 20

2:10pm Eastern (note different time)

Vijay Higgins

Central elements in the SL(d) skein algebra

October 27

2pm Eastern

Abhishek Mallick

Detecting corks

November 3

2pm Eastern

Christine Lee

Stable Khovanov homology of torus links and volume

November 10

2pm Eastern  (double header pt 1)

Ying Hu

Stir fry Homeo_+(S^1) representations from pseudo-Anosov flows 

November 10

4pm Eastern  (double header pt 2)

Francisco Arana Herrera

Arithmetic v/s geometric complexity of simple closed geodesics
November 17

2:10 pm Eastern (note different time)

Antonio Alfieri

Involutions and the Chern-Simons filtration in instanton Floer homology

December 1

2pm Eastern

Bulent Tosun

Embedding questions for 3-manifolds

December 8

2pm Eastern

Diana Hubbard

Twisting in small and big mapping class groups




Name: Liam Watson

Date: September 8

TitleKhovanov multicurves are linear 

Abstract: For a given invariant the geography problem asks for a characterization of values the invariant attains. For example, it is well understood which Laurent polynomials arise as the Alexander polynomial of a knot. By contrast, very little is known about which values the Jones polynomial takes. And the situation is at least as bad for Khovanov homology. So it is a little surprising that the Khovanov homology of a tangle, which can be framed in terms of immersed curves in the 4-punctured sphere, satisfies rather strict geography—the invariants are “linear”.  My talk, which is based on joint work with Artem Kotelskiy and Claudius Zibrowius, will explain what “linear” means, and tell part of the story of how this comes up. 


Name: Anna Parlak

Date: September 15

TitleFaces of the Thurston norm ball dynamically represented by multiple distinct flows

Abstract: A pseudo-Anosov flow on a closed 3-manifold dynamically represents a face F of the Thurston norm ball if the cone on F is dual to the cone spanned by homology classes of closed orbits of the flow. Fried showed that for every fibered face of the Thurston norm ball there is a unique, up to isotopy and reparameterization, flow which dynamically represents the face.  Mosher found sufficient conditions on a non-circular flow to dynamically represent a non-fibered face, but the problem of the existence and uniqueness of the flow for every non-fibered face was unresolved. I will outline how to show that a non-fibered face can be in fact dynamically represented by multiple topologically inequivalent flows, and discuss how two distinct flows representing the same face may be related.

Name: Matt Hogancamp

Date: September 22

Title: Khovanov homology and handleslides

Abstract: If K is a framed knot and X is a link in the solid torus, then the satellite operation produces a link K(X), obtained by embedding X (the "pattern") into a tubular neighborhood of K (the  "companion"). In the context of Khovanov homology, the patterns can be thought of as objects of a category called the annular Bar-Natan category (ABN), and the satellite operation defines a functor from ABN to bigraded vector spaces, sending a pattern X to Kh(K(X)). In this talk I will discuss joint work with Dave Rose and Paul Wedrich, in which we construct an object Ω  (which we call a "Kirby color") in ABN such that Kh(K(Ω)) is invariant under handleslides. As I will explain, the object Ω encodes the Manolescu-Neithalath 2-handle formula for the sl(2) skein lasagna modules (which was inspirational for our work). Time permitting, I will discuss an intriguing description of the Kirby object in terms of some special braids (positive braid lifts of n-cycles) that we speculate may be more amenable for computation.


Name: Inanc Baykur

Date:October 6

TitleExotic 4-manifolds with signature zero

Abstract:I'll tell about our recent construction of small symplectic 4-manifolds with signature zero, which include the smallest closed simply-connected exotic 4-manifolds with signature zero known to date. Our novel examples are derived from explicit Lefschetz fibrations, with spin and non-spin monodromies. Joint work with N. Hamada.


Name: Ian Agol

Date:October 13

TitleRibbon concordance of knots is a partial order

Abstract:We will discuss the result in the title, answering a question of Gordon. The proof makes use of representation varieties of the knot group to SO(N). If there is time, we will discuss the prospect of answering some other questions from Gordon’s paper about ribbon concordance. 

Name: Vijay Higgins

Date:October 20

Title: Central elements in the SL(d) skein algebra

Abstract: The skein algebra of a surface is spanned by links in the thickened surface, subject to skein relations which diagrammatically encode the data of a quantum group. The multiplication in the algebra is induced by stacking links in the thickened surface. This product is generally noncommutative. When the quantum parameter q is generic, the center of the skein algebra is essentially trivial. However, when q is a root of unity, interesting central elements arise. When the quantum group is quantum SL(2), the work of Bonahon-Wong shows that these central elements can be obtained by a topological operation of threading Chebyshev polynomials along knots. In this talk, I will discuss joint work with F. Bonahon in which we use analogous multi-variable 'threading' polynomials to obtain central elements in higher rank SL(d) skein algebras. For a closed surface, we conjecture these elements generate the center of the skein algebra.


Name: Abhishek Mallick

Date:October 27

TitleDetecting corks

Abstract: Corks are fundamental objects in the study of exotic smooth 4-manifolds. In this talk we will describe various methods for detecting corks. We will then describe some new examples of corks which partially address a question posed by Gompf. Part of this talk is work in progress with Irving Dai and Ian Zemke.


Name: Christine Lee

Date:November 3

Title: Stable Khovanov homology of torus links and volume

Abstract: Let T(n, k) denote the (n, k)-torus braid. It is well known that the Jones polynomial and the Khovanov homology of torus links stabilize as k → ∞ by the work of Champanekar-Kofman and Stosic. In particular, Rozanksy showed that the stable Khovanov homology of torus links exists as the direct limit of the Khovanov homology of T(n, k)-torus links, and the stable Khovanov homology recovers the categorification of the Jones-Wenzl projector. We show that the categorification of the Khovanov homology of a link stabilizes under twisting as a categorial analogue of the result by Champanekar-Kofman, extending the results by Stosic and Rozansky. Since the Jones-Wenzl projector can be used to define the colored Jones polynomial, we will discuss potential relationship between the stable invariant to the hyperbolic volume of a knot in the spirit of the volume conjecture.


Name: Ying Hu

Date:November 10 

TitleStir fry Homeo_+(S^1) representations from pseudo-Anosov flows 

Abstract: A total linear order on a non-trivial group G is a left-order if it’s invariant under group left-multiplication. A result of Boyer, Rolfsen and Wiest shows that a 3-manifold group has a left-order if and only if it admits a non-trivial representation into Homeo_+(S^1) with zero Euler class. Foliations, laminations and flows on 3-manifolds often give rise to natural non-trivial Homeo_+(S^1)-representations of the fundamental groups, which have proven to be extremely useful in studying the left-orderability of 3-manifold groups.  In this talk, we will present a recipe of stir frying these Homeo_+(S^1)-representations. Our operation generalizes a previously known ``flipping'' operation introduced by Calegari and Dunfield. As a consequence, we constructed a surprisingly large number of new Homeo_+(S^1)-representations of the link groups. We then use these newly obtained representations to prove the left-orderablity of cyclic branched covers of links associated with any epimorphisms to Z_n.  This is joint work with Steve Boyer and Cameron Gordon.


Name: Francisco Arana Herrera 

Date:November 10

TitleArithmetic v/s geometric complexity of simple closed geodesics

Abstract: How homologically complicated are long simple closed geodesics on hyperbolic surfaces? We provide an answer to this question which is surprisingly different from the well studied case of general primitive closed geodesics. We explain the relation between these questions and mixing limit theorems for the Kontsevich-Zorich cocycle. Parts of this talk are joint work in progress with Pouya Honaryar and other parts are joint work with Giovanni Forni. Time permitting we discuss several open questions in the field.

Name: Antonio Alfieri 

Date:November 17

TitleInvolutions and the Chern-Simons filtration in instanton Floer homology

Abstract: In a recent paper in collaboration with Dai, Mallick, and Taniguchi we developed an invariant based on instanton Floer homology aimed at studying strong corks and equivariant bounding. Our construction utilizes the Chern-Simons filtration and is qualitatively different from previous Floer-theoretic methods used to address these questions. I will discuss various topological applications of these methods and outline the main ideas behind this machinery.  


Name: Bulent Tosun

Date: December 1

TitleEmbedding questions for 3-manifolds

Abstract: In this talk I will discuss 3-manifold embedding questions into 4-dimensional manifolds from the perspectives of smooth topology as well as symplectic and complex geometry. The talk will have two parts: In the first part the focus will be on embedding questions into 4-dimensional Euclidean space (R^4). Given a closed, orientable 3-manifold Y, it is of great interest but often a difficult problem to determine whether Y may be smoothly embedded in R^4. This is the case even for integer homology spheres (where usual obstructions coming from homology disappear), and restricting to special classes such as Seifert manifolds, the problem is open in general. On the other hand, under additional geometric considerations coming from symplectic geometry (such as hypersurfaces of contact type in R^4)and complex geometry (such as the boundaries of pseudo-convex or rationally convex domains in complex Euclidean space C^2), the problems become tractable and in certain cases a uniform answer is possible. For example, recent work shows no correctly oriented Seifert homology sphere admits an embedding as a hypersurface of contact type in R^4. In the second part, I will consider general closed 4-manifolds as target manifolds and mention some recent work in progress. I will provide further context and motivations for these results, and give some details of the proofs.  


Name: Diana Hubbard

Date: December 8

TitleTwisting in small and big mapping class groups

Abstract: For a finite type surface with boundary, the fractional Dehn twist coefficient is a rational number that measures, informally, the amount of twisting a mapping class of the surface effects about a chosen boundary component. This quantity has connections to classical knot theory, open book decompositions, and contact geometry. In this talk I will generalize this notion to the infinite type setting by characterizing it as the unique quasimorphism from the mapping class group of the surface to the real numbers with certain properties. Unlike in the finite type setting, for certain infinite type surfaces this quasimorphism has image all of R. To see this I will explain a new construction of maps with irrational rotation behavior. I will discuss how this work may be of use in the search for "Nielsen-Thurston"-type classification results for infinite type surfaces. This work is joint with Peter Feller and Hannah Turner.


Other relevant information.

Previous semesters:

Spring 2023Fall 2022Spring 2022Fall 2021, Spring 2021, Fall 2020Spring 2020Fall 2019Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

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