# Columbia Geometric Topology Seminar

Fall 2019

Organizer: Nick Salter.
The GT seminar meets on Fridays at 2:00pm Fridays in room 520. We also have an overflow room 622 from 11 to 1 Fridays for additional talks.

Other area seminars. Our e-mail list. Archive of previous semesters

## Fall 2019

Date Speaker Title
Sept 13 Indira Chatterji Group actions on median spaces and generalizations
Sept 20 Ty Ghaswala Promoting circle actions to actions on the real line
Sept 27 Danny Calegari Big mapping class groups and rigidity of the simple circle
Oct 4 Khalid Bou-Rabee Quantifying residual finiteness
Oct 11 Dusa McDuff Counting curves with tangency constraints in the complex projective plane
Oct 18 Margaret Nichols Taut sutured handlebodies as twisted homology products
Oct 25 Francesco Lin Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants
Nov 1 Abdul Zalloum CAT(0) groups are determined by sublinear boundaries
Nov 8 Nick Vlamis
Nov 15 Peter Lambert-Cole
Nov 22 Diana Hubbard
Dec 6 Jingyin Huang
Dec 13 Maggie Miller

## Abstracts

Abdul Zalloum, Queens University

To each hyperbolic space, one can associate a space at infinity called the Gromov’s boundary. Gromov showed that a quasi-isometry between two hyperbolic spaces induces a homeomorphism on their boundaries. For a CAT(0) space, one can also assign a space at infinity called the visual boundary but that is no longer a quasi-isometry invariant. Several attempts have been made to circumvent the problem, most recent of which is work by Qing and Rafi. They introduce the notion of a ”sublinear contracting boundary” of a CAT(0) space and they show that a quasi-isometry between two CAT(0) spaces induces a homeomorphism between their sublinear boundaries. We investigate when the converse holds: Given a homeomorphism between two sublinear boundaries of CAT(0) spaces, when is it induced by a quasi isometry? We show that a homeomorphism f between two cocompact CAT(0) spaces f:X-->Y is induced by a quasi-isometry if and only if f is stable and Morse quasi-Mobius.  In this talk, I will define all the objects above and give a sketch for the proof. This is joint work with Yulan Qing.

Francesco Lin, Columbia

October 25, 2019
Title
Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants
Abstract:
We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic 4-manifolds contain L-spaces as hypersurfaces. This is joint work with Ian Agol.

Reference:https://arxiv.org/abs/1812.06536

Margaret Nichols, SUNY Buffalo

October 18, 2019
Title
Taut sutured handlebodies as twisted homology products
Abstract:
A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’.

One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such?

We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples that, when restricted to solvable representations, the twisting representation cannot be ‘too simple’.

Dusa McDuff, Columbia

October 11, 2019
Title
Counting curves with tangency constraints in the complex projective plane
Abstract:
The attempt to prove the stabilized ellipsoidal symplectic embedding problem has lead to some very interesting questions about the behavior of genus zero pseudo-holomorphic curves in $\C P^2$.  In particular, it seems that one can generalize the Caporaso--Harris recursion formula that counts curves tangent to a global divisor to cases where one also allows tangencies to local divisors.  I will explain the embedding problem, the relevance of curve counts to it, and then explain some recent joint work with Kyler Siegel that develops new ways to perform these counts.   The talk will assume no special knowledge of symplectic geometry.

Khalid Bou-Rabee, CUNY

October 4, 2019
Title
Quantifying residual finiteness
Abstract:
The theory of quantifying residual finiteness assigns, to each finitely generated group, an invariant that indicates how well-approximated the group is by its finite quotients. We introduce this theory and survey the current state of the subject. There will be a strong emphasis on examples, open questions, and connections to other subjects.

Danny Calegari, University of Chicago

September 27, 2019
Title: Big mapping class groups and rigidity of the simple circle
Abstract:
Let G denote the mapping class group of the plane minus a Cantor set. We show that every action of G on the circle  is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle. This is joint work with Lvzhou (Joe) Chen.

Ty Ghaswala, University of Manitoba

September 20, 2019
TitlePromoting circle actions to actions on the real line
Abstract
Circularly-orderable and left-orderable groups play an important, and sometimes surprising, role in low-dimensional topology and geometry. For example, these combinatorial conditions completely characterize when a countable group acts on a 1-manifold. Through the so-called L-space conjecture, left-orderability of the fundamental group of a rational homology 3-sphere is connected to the existence of taut foliations and properties of its Heegaard Floer homology. I will present new necessary and sufficient conditions for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. This raises a plethora of intriguing questions, especially in the case when the group is the fundamental group of a manifold.

This is joint work with Jason Bell and Adam Clay.

Indira Chatterji, CNRS
September 13, 2019
TitleGroup actions on median spaces and generalizations
Abstract
Median spaces are a natural generalization of R-trees and CAT(0) cubical complexes. I will define the notion, discuss the context and show that the fundamental group of a compact hyperbolic manifold acts properly and cocompactly on a median space.