Seminars and Colloquia by Series

On obstructing Lagrangian concordance

Series
Geometry Topology Seminar
Time
Monday, February 20, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Angela Wu – Lousiana State University –

Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. Interestingly, Lagrangian concordance is, unlike smooth concordance, not a symmetric relation. In this talk, I'll discuss various strategies that can be used to obstruct Lagrangian concordance, from basic invariants of Legendrian knots, to the Chekanov-Eliashberg DGA, to building new obstructions from Weinstein cobordisms.

Generalized square knots, homotopy 4-spheres, and balanced presentations

Series
Geometry Topology Seminar
Time
Monday, February 13, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 303)
Speaker
Jeff Meier – Western Washington University

We will describe an elegant construction of potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture whose input is a fibered, homotopy-ribbon knot in the 3-sphere. The construction also produces links that are potential counterexamples to the Generalized Property R Conjecture, as well as balanced presentations of the trivial group that are potential counterexamples to the Andrews-Curtis Conjecture. We will then turn our attention to generalized square knots (connected sums of torus knots with their mirrors), which provide a setting where the potential counterexamples mentioned above can be explicitly understood. Here, we show that the constructed 4-manifolds are diffeomorphic to the 4-sphere; but the potential counterexamples to the other conjectures persist. In particular, we present a new, large family of geometrically motivated balanced presentations of the trivial group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy and isotopy rel-boundary. This talk is based on joint work with Alex Zupan.

Handle numbers of nearly fibered knots

Series
Geometry Topology Seminar
Time
Monday, February 13, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
University of Georgia & Zoom
Speaker
Ken Baker – University of Miami

In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2. Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4.  Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces. This is joint work in progress with Fabiola Manjarrez-Gutierrez.

Distinguishing hyperbolic knots using finite quotients

Series
Geometry Topology Seminar
Time
Monday, February 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Tam Cheetham-West – Rice University –

The fundamental groups of knot complements have lots of finite quotients. We give a criterion for a hyperbolic knot in the three-sphere to be distinguished (up to isotopy and mirroring) from every other knot in the three-sphere by the set of finite quotients of its fundamental group, and we use this criterion as well as recent work of Baldwin-Sivek to show that there are infinitely many hyperbolic knots distinguished (up to isotopy and mirroring) by finite quotients. 

Higher Complex Structures and Hitchin Components

Series
Geometry Topology Seminar
Time
Monday, January 30, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Nolte – Rice/Georgia Tech

A source of richness in Teichmüller theory is that Teichmüller spaces have descriptions both in terms of group representations and in terms of hyperbolic structures and complex structures. A program in higher-rank Teichmüller theory is to understand to what extent there are analogous geometric interpretations of Hitchin components. In this talk, we will give a natural description of the SL(3,R) Hitchin component in terms of higher complex structures as first described by Fock and Thomas. Along the way, we will describe higher complex structures in terms of jets and discuss intrinsic structural features of Fock-Thomas spaces.

On the homology of Torelli groups

Series
Geometry Topology Seminar
Time
Monday, January 23, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan Minahan – Georgia Institute of Technology

The Torelli group of a surface is a natural yet mysterious subgroup of the mapping class group.  We will discuss a few recent results about finiteness properties of the Torelli group, as well as a result about the cohomological dimension of the Johnson filtration.  

 

A Tale of Two Theorems of Thurston

Series
Geometry Topology Seminar
Time
Monday, January 9, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan Margalit – Georgia Institute of Technology –

In the 20th century, Thurston proved two classification theorems, one for surface homeomorphisms and one for branched covers of surfaces.  While the theorems have long been understood to be analogous, we will present new work with Belk and Winarski showing that the two theorems are in fact special cases of one Ubertheorem.  We will also discuss joint work with Belk, Lanier, Strenner, Taylor, Winarski, and Yurttas on algorithmic aspects of Thurston’s theorem.  This talk is meant to be accessible to a wide audience.

d-Pleated surfaces and their coordinates

Series
Geometry Topology Seminar
Time
Monday, December 12, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Giuseppe Martone – Yale
Thurston introduced pleated surfaces as a powerful tool to study hyperbolic 3-manifolds. An abstract pleated surface is a representation of the fundamental group of a hyperbolic surface into the Lie group PSL(2,C) of orientation preserving isometries of hyperbolic 3-space together with an equivariant map from the hyperbolic plane into hyperbolic 3-space which satisfies additional properties.
 
In this talk, we introduce a notion of d-pleated surface for representations into PSL(d,C) which is motivated by the theory of Anosov representations. In addition, we give a holomorphic parametrization of the space of d-pleated surfaces via cocyclic pairs, thus generalizing a result of Bonahon.

This talk is based on joint work with Sara Maloni, Filippo Mazzoli and Tengren Zhang.
 

Multisections, the pants complex, and Weinstein manifolds

Series
Geometry Topology Seminar
Time
Monday, December 5, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gabriel Islambouli – UC Davis –

We introduce a decomposition of a 4-manifold called a multisection, which is a mild generalization of a trisection. We show that these correspond to loops in the pants complex and provide an equivalence between closed smooth 4-manifolds and loops in the pants complex up to certain moves. In another direction, we will consider multisections with boundary and show that these can be made compatible with a Weinstein structure, so that any Weinstein 4-manifold can be presented as a collection of curves on a surface.

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