Columbia Geometric Topology Seminar

Fall 2022


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Organizer: Daniele AlessandriniSiddhi Krishna, Francesco Lin

The GT seminar typically meets on Fridays at 2:00pm Eastern time in Room 520, Mathematics Department, Columbia University. It will also be live-streamed over Zoom.  

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Fall 2022

Date Time (Eastern) Speaker Title

September 16


Amina Abdurrahman

A global cohomological formula for Reidemeister torsion

September 23


Jean Pierre Mutanguha

Canonical forms for free group automorphisms

September 30


Marco Marengon

Relative genus bounds in indefinite 4-manifolds

October 7


no speaker

No seminar (Krichever Conference)

October 14


Marta Magnani

Parametrizing the space of maximal representations

October 21


Sam Taylor

Endperiodic maps via pseudo-Anosov flows

October 28


Funda Gultepe

Curves, hexagons and geometry of surfaces

November 4 

Double Header Pt 1 @ 2pm  

Peter Feller

On the length of knots on a Heegaard surface of a 3-manifold

November 4

Double Header Pt 2 @ 4:45pm  

John Baldwin

Floer homology and non-fibered knots

November 11


Thang Nguyen

Marked length spectrum rigidity for relatively hyperbolic groups

November 18


Jane Wang

The topology of the moduli space of dilation surfaces

November 25


no speaker

Happy Thanksgiving!

December 2


Katie Mann

Classifying Anosov flows on 3-manifolds

December 9


Gary Guth

Satellites, Stabilizations, and Exotic Surfaces



September 16: Amina Abdurrahman (Stony Brook)

Title: A global cohomological formula for Reidemeister torsion
Abstract: We give a global cohomological formula for Reidemeister torsion of a 3-manifold together with a symplectic local system. This can be considered as the topological analogue of a number-theoretic formula generalizing a result of Deligne in the 70s about local espilon factors. We plan to tell the topological story and touch on some of the related ideas in topology and number theory.

September 23: Jean Pierre Mutanguha (Princeton)

Title: Canonical forms for free group automorphisms
Abstract: The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

September 30: Marco Marengon (Renyi Institute)

Title: Relative genus bounds in indefinite 4-manifolds
Abstract: Given a closed 4-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X − int(B^4), with boundary a given knot K in the 3-sphere. We give several methods to bound the genus of such surfaces in a fixed homology class. Our techniques include adjunction inequalities from Heegaard Floer homology and the Bauer-Furuta invariants, and the 10/8 theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disc) in a 4-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds. This is joint work with Ciprian Manolescu and Lisa Piccirillo.

October 14: Marta Magnani (Heidelberg)

Title: Parametrizing the space of maximal representations
Abstract: Higher rank Teichmüller theory was developed as a generalization of classical Teichmüller theory and is concerned with the study of representations of fundamental groups of oriented surface S of negative Euler characteristic into simple real Lie groups G of higher rank. In the talk we will introduce Higher rank Teichmüller theory with particular attention to maximal representations. We will then introduce the Siegel space and discuss how, given a hyperbolic surface with boundary, one can parametrize the space of maximal representations from the fundamental group of the surface into PSp(2n,R).


October 21: Sam Taylor (Temple)

Title: Endperiodic maps via pseudo-Anosov flows
Abstract: We show that every atoroidal, endperiodic map of an infinite-type surface is isotopic to a homeomorphism that is naturally the first return map of a pseudo-Anosov suspension flow on a fibered manifold. Morally, these maps are all obtained by “spinning” fibers around a surfaces in the boundary of the fibered cone. The structure associated to these spun pseudo-Anosov maps allows for several applications. These include defining and characterizing stretch factors of endperiodic maps, relating Cantwell—Conlon foliation cones to Thurston’s fibered cones, and defining a convex entropy function on these cones that extends log(stretch factor). This is joint work with Michael Landry and Yair Minsky.


October 28: Funda Gultepe (Toledo)

Title: Curves, hexagons and geometry of surfaces
Abstract: In this talk, we define and study graphs associated to hexagon decompositions of surfaces by curves and arcs and relate them to pants graph and the mapping class group. We will also give an estimate on the diameter of the moduli space of hexagon decompositions on a surface. This is a joint work with Hugo Parlier.
November 4: Peter Feller (ETH Zurich)
Title: On the length of knots on a Heegaard surface of a 3-manifold.
Abstract: 3-manifold theory has expanded its tool box in recent decades: topological, (Floer and quantum) homological, and geometrical methods all have been employed with success. However, often the relation between these different approaches remains mysterious.

In this talk we explore connections between the topology and the geometry of 3-manifolds by using Heegaard-splittings (topology) of a 3-manifold to describe hyperbolic structures (geometry) on it. More concretely, for a knot K that lies on a Heegaard surface of a closed oriented connected 3-manifold M, we describe a sufficient condition for M to carry a hyperbolic structure. Furthermore, whenever our criterion applies, we determine the length of K up to a multiplicative constant.

Upshot of our approach: there is NO Ricci-flow machine running in the background. Instead, the motor behind what we do is an effective version of Thurston's hyperbolic Dehn surgery. Applications include a Ricci-flow free proof of Mather's result that random 3-manifolds (in the sense of Dunfield-Thurston) are hyperbolic, and bounds on the diameter and injectivity radius of a random 3-manifold.
November 4: John Baldwin (Boston College)
Title: Floer homology and non-fibered knots
Abstract: A fundamental question for any knot invariant asks which knots it detects. For example, it is a famous open question whether the Jones polynomial detects the unknot. The detection question for knot Floer homology and the Khovanov-Rozansky link homology theories has received a lot of attention of the past two decades, culminating in proofs that these theories detect six knots: the unknot, the trefoils, the figure eight, and the cinquefoils. Crucial in each of these detection results (save for that of the unknot) is that the knot in question is fibered. I'll discuss recent work with Sivek in which we show for the first time that knot Floer homology and Khovanov homology can also detect non-fibered knots, and that HOMFLY homology can in fact detect infinitely many knots. 
November 4: Thang Nguyen (Florida State)
Title: Marked length spectrum rigidity for relatively hyperbolic groups
Abstract: Burns and Katok asked, among homeomorphic manifolds of negative sectional curvature, whether the lengths of the family of marked geodesic loops determine the geometry of a manifold. I will state a coarse version of this question for finitely generated groups. After going over some previously known results, we'll focus our attention on the case of relatively hyperbolic groups. This is based on a joint work with Shi Wang.
November 18: Jane Wang (Maine)
Title: The topology of the moduli space of dilation surfaces
Abstract: Translation surfaces are geometric objects that can be defined as a collection of polygons with sides identified in parallel opposite pairs by translation, or as a Riemann surface together with a holomorphic one-form. If we generalize slightly and allow for polygons with sides identified by both translation and dilation, we get a new family of objects called dilation surfaces. While translation surfaces are well-studied, much less is known about dynamics on dilation surfaces and their moduli spaces. In this talk, we will survey recent progress in understanding the topology of moduli spaces of dilation surfaces, including realizing the fundamental groups of these moduli spaces as certain subgroups of the mapping class group. This talk represents joint work with Paul Apisa and Matt Bainbridge.
December 2: Katie Mann (Cornell)
Title: Classifying Anosov flows on 3-manifolds
Abstract: Anosov flows are rich examples of dynamical systems, they include the geodesic flows on unit tangent bundles of hyperbolic surfaces, and many other examples.  This talk is about how dynamics, geometry and topology interact in dimension 3 via some longstanding open questions:  Which 3-manifolds support Anosov flows?  Which 3-manifolds support many topologically distinct Anosov flows?  What invariants can be used to distinguish them?  I will describe some of the state of the art, and recent work with Thomas Barthelmé, Steven Frankel, and Sergio Fenley that provides new topological invariants towards this classification problem.  


December 9: Gary Guth (Oregon)
Title: Satellites, Stabilizations, and Exotic Surfaces
Abstract: A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable". For example, in thinking about the four-dimensional h-cobordism theorem, Wall proved that simply connected, exotic four-manifolds always become smoothly equivalent after applying a suitable stabilization operation enough times. Similarly, Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. The question remains, "how many stabilizations are necessary, and is one always enough?" By considering certain satellite operations, we provide an answer to this question in the case of exotic surfaces with boundary



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