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 | Topology of (big) mapping class groups |

Nov 15 | Kyle Hayden | Knots, spines, and exotic 4-manifolds |

Nov 22 | Diana Hubbard | Right-veering open books and the Upsilon invariant |

Dec 6 | Jingyin Huang | The Helly geometry of some Garside and Artin groups |

Dec 13 | Maggie Miller | Light bulbs in 4-manifolds |

## Abstracts

**Maggie Miller, Princeton**

**NOTE: TALK IS IN ROOM 622**

December 13, 2019

**Title**: Light bulbs in 4-manifolds

**Abstract**: In 2017, Gabai proved the light bulb theorem, showing that if $R$ and $R'$ are 2-spheres homotopically embedded in a 4-manifold with a common dual, then with some condition on 2-torsion in $\pi_1(X)$ one can conclude that $R$ and $R'$ are smoothly isotopic. Schwartz gave examples showing that this 2-torsion condition is necessary, and Schneiderman and Teichner then obstructed the isotopy whenever this condition fails, yielding a complete solution to when light bulbs are isotopic. I weakened the hypothesis on $R$ and $R'$ by allowing $R'$ not to have a dual and showed that the spheres are still smoothly concordant.

**References:**https://arxiv.org/abs/1903.03055

**Jingyin Huang, Ohio State**

December 6, 2019

**Title**: The Helly geometry of some Garside and Artin groups

**Abstract**: Garside groups and Artin groups are two generalizations of braid groups. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, hence equip these groups with a particular nonpositive-curvature-like (NPC) structure. Such structure shares many properties of a CAT(0) structure and has some additional combinatorial flavor. We shall explain this NPC structure in more detail and discuss new results on the topology and geometry of these groups which are immediate consequences of such structure. This is joint work with D. Osajda.

**References:** https://arxiv.org/abs/1904.09060

**Diana Hubbard, CUNY**

November 22, 2019

**Title**: Right-veering open books and the Upsilon invariant

**Abstract**: Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

**Kyle Hayden, Columbia**

November 15, 2019

**Title**: Knots, spines, and exotic 4-manifolds

**Abstract**: A "knot trace" is a simple 4-manifold obtained by attaching a thickened disk (i.e. a 2-handle) to the 4-ball along a knot in its boundary. These 4-manifolds arise in many contexts, including the construction of exotic smooth structures on 4-space. After a quick survey of this history, I'll use knot traces to present a simple alternative solution to a classical question about piecewise linear surfaces in 4-manifolds (Problem 4.25 on Kirby's list, recently resolved by Levine and Lidman). The underlying construction provides pairs of homeomorphic 4-manifolds X and X' such that X is a knot trace and yet X' is not diffeomorphic to any knot trace, including X. These "not knot traces" X' exhibit a variety of other curious phenomena; I'll briefly mention some applications to the study of handle decompositions of simply-connected 4-manifolds, Stein fillings of contact 3-manifolds, and knot concordance. This is joint work in progress with Lisa Piccirillo.

**Nick Vlamis, CUNY**

November 8, 2019

**Title**: Topology of (big) mapping class groups

**Abstract**: Mapping class groups inherit a natural topology from the compact-open topology on homeomorphism groups. When the underlying surface is of infinite type, this topology is no longer discrete, which allows us to study these mapping class groups from the perspective of topological group theory. In the talk, I will discuss two results, both of which highlight applications of this topological group theoretic perspective.

**References:**https://arxiv.org/abs/1711.03132

**Abdul Zalloum, Queens University**

November 1, 2019

**Title**: CAT(0) groups are determined by sublinear boundaries

**Abstract**: 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’.

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

**Title**: Promoting 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

**Title**: Group 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.

# Other relevant information.

## Previous semesters:

Spring 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.## Other area seminars.

- Columbia Symplectic Geometry/Gauge Theory Seminar
- All Columbia Math Dept Seminars
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.