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Columbia Geometric Topology SeminarFall 2023 

Organizers: Ross Akhmechet, Siddhi Krishna, Francesco 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.
Other area seminars. Our email list. Archive of previous semesters
Date  Time (Eastern)  Speaker  Title 

September 8 
2pm Eastern 
Liam Watson 
Khovanov multicurves are linear 
September 15 
2pm Eastern 
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 
n.a. 
no seminar today! 
October 6 
2pm Eastern 
Exotic 4manifolds with signat 

October 13 
4:30pm Eastern in Room 312 
Ribbon concordance of knots is a partial order 

October 20 
2:10pm Eastern (note different time) 
Central elements in the SL(d) skein algebra 

October 27 
2pm Eastern  Detecting corks  
November 3 
2pm Eastern 
Stable Khovanov homology of torus links and volume 

November 10 
2pm Eastern (double header pt 1) 
Stir fry Homeo_+(S^1) representations from pseudoAnosov flows  
November 10 
4pm Eastern (double header pt 2) 
Arithmetic v/s geometric complexity of simple closed geodesics  
November 17 
2:10 pm Eastern (note different time) 
Involutions and the ChernSimons filtration in instanton Floer homology  
December 1 
2pm Eastern 
Embedding questions for 3manifolds 

December 8 
2pm Eastern 
Twisting in small and big mapping class groups 
Name: Liam Watson
Title: Khovanov 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 4punctured 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
Title: Faces of the Thurston norm ball dynamically represented by multiple distinct flows
Abstract: A pseudoAnosov flow on a closed 3manifold 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 noncircular flow to dynamically represent a nonfibered face, but the problem of the existence and uniqueness of the flow for every nonfibered face was unresolved. I will outline how to show that a nonfibered 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
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 BarNatan 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 ManolescuNeithalath 2handle 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 ncycles) that we speculate may be more amenable for computation.
Name: Inanc Baykur
Title: Exotic 4manifolds with signat
Abstract:I'll tell about our recent construction of small symplectic 4manifolds with si
Name: Ian Agol
Title: Ribbon 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.
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 BonahonWong 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 multivariable '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
Title: Detecting corks
Abstract: Corks are fundamental objects in the study of exotic smooth 4manifolds. 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
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 ChampanekarKofman 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 JonesWenzl projector. We show that the categorification of the Khovanov homology of a link stabilizes under twisting as a categorial analogue of the result by ChampanekarKofman, extending the results by Stosic and Rozansky. Since the JonesWenzl 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
Title: Stir fry Homeo_+(S^1) representations from pseudoAnosov flows
Abstract: A total linear order on a nontrivial group G is a leftorder if it’s invariant under group leftmultiplication. A result of Boyer, Rolfsen and Wiest shows that a 3manifold group has a leftorder if and only if it admits a nontrivial representation into Homeo_+(S^1) with zero Euler class. Foliations, laminations and flows on 3manifolds often give rise to natural nontrivial Homeo_+(S^1)representations of the fundamental groups, which have proven to be extremely useful in studying the leftorderability of 3manifold 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 leftorderablity 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
Title: Arithmetic 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 KontsevichZorich 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
Title: Involutions and the ChernSimons 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 ChernSimons filtration and is qualitatively different from previous Floertheoretic 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
Title: Embedding questions for 3manifolds
Abstract: In this talk I will discuss 3manifold embedding questions into 4dimensional 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 4dimensional Euclidean space (R^4). Given a closed, orientable 3manifold 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 pseudoconvex 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 4manifolds 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
Title: Twisting 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 "NielsenThurston"type classification results for infinite type surfaces. This work is joint with Peter Feller and Hannah Turner.