BYU Seal Calvin College Seal   

The 39th Annual

(Online) Workshop in
Geometric Topology

June 6 - 8, 2022

   Colorado College Seal TCU Seal University of Wisconsin-Milwaukee Seal

Workshop Information

Jessica Purcell This year's workshop will be completely online. Registrants will be sent connection details prior to the Workshop.

Deadlines

Submission of contributed talks, including title and abstract: May 6

Principal Speaker

The featured speaker is Jessica Purcell of Monash University, who will give a series of three one-hour lectures on using hyperbolic geometry as a tool to study 3-manifolds and knot theory. An abstract can be found below.

Contributed Talks

The workshop will also feature contributed talks of 20 minutes. Contributed talks need not be directly related to the topic of the principal lectures. If you wish to apply to give a contributed talk please submit a title and abstract on the registration form or send directly to Greg Friedman (send email). Talks will be selected in a way that provides a balanced collection of topics and respects the historical traditions of the workshop. Earlier responses may be given some preference.Applicants will be notified whether their talk has been accepted by May 23.

Abstract of the Main Lectures

For over four decades, hyperbolic geometry has been used as a tool to study and classify 3-manifolds. Hyperbolic 3-manifolds are known to satisfy important rigidity results, so that hyperbolic structures are unique or well-behaved. Thus hyperbolic invariants, such as volume, areas of embedded surfaces, and lengths of short geodesics, become topological 3-manifold invariants. However, many of the 3-manifolds that arise in geometric topology, such as examplesfrom knot theory, dynamics, or piecewise linear topology, are constructed in ways that do not obviously reflect their geometry. Given a 3-manifold with a combinatorialor topological description, it can often be difficult to relate that description to hyperbolic geometry of the manifold. In my talks, I will survey some of the ways of relating hyperbolic geometry to combinatorics and topology in dimension three, and some applications.

Day 1: Hyperbolic knots and alternating knots
Hyperbolic geometry has been used since around the mid-1970s to study knot theory, but it can be difficult to relate geometry of knots to a diagram of a knot. However, many results from the 1980s and beyond show that alternating knots have diagrams that seem to more closely reflect their geometric properties than other general knots. In the first part, I値l introduce hyperbolic geometry and knots, and survey some classical results. In the second part, I値l describe results specific to alternating knots, and explain some of the tools that are useful to prove results in the alternating case. In the last part of the talk, I値l explain new methods to extend these tools, to extract geometry from much larger families of knots, discussing some of my recent joint work with Josh Howie and with Effie Kalfagianni.

Day 2: Fully augmented links and circle packings
Fully augmented links form a family of hyperbolic links that are a playground for hands-on hyperbolic geometry. In the first part of the talk, I値l define the links and show how to determine their hyperbolic geometry explicitly. I will show how such links are determined by circle packings of the sphere, and vice versa. The theory of circle packings extends beyond spheres. I値l then explain new work, building fully augmented links on higher genus surfaces. In the last part of the talk, I値l give some applications to geometric convergence of knots and links. The latter part will describe recent joint work with Urs Fuchs and John Stewart.

Day 3: Knots in infinite volume 3-manifolds
Classically, knots have been studied in the 3-sphere. However, many examples of knots arising in applications lie in broader classes of 3-manifolds. These include virtual knots, which lie within thickened surfaces, and periodic orbits of dynamical systems. Not all such manifolds have finite hyperbolic volume. However, the last two decades have brought new tools to bear on hyperbolic geometry of infinite volume 3-manifolds. In the first part of the talk, I値l survey some of those results. I値l then discuss techniques to mix infinite-volume tools with finite-volume ones, to give further geometric information on knots in general 3-manifolds. This includes a 6-Theorem for tame 3-manifolds, and additional results controlling geometry changes under Dehn filling. The latter part of the talk is joint work with David Futer and Saul Schleimer.

Short Biography of Jessica Purcell

Jessica Purcell is a professor of mathematics at Monash University in Australia. She works on problems in the overlap of hyperbolic geometry, 3-manifolds, and knot theory. Prior to arriving at Monash in 2015, she obtained a PhD in Mathematics from Stanford University, held postdoctoral positions at the University of Texas in Austin and the University of Oxford, and worked at Brigham Young University from 2007 to 2015. She has received several awards and grants for her research, including a Future Fellowship from the Australian Research Council from 2017 to 2021, a Von Neumann Fellowship at the Institute for Advanced Study in 2015, a Sloan Research Fellowship in 2011, and a CAREER award from the National Science Foundation of the USA from 2013 to 2015. Outside of mathematical research, Jessica recently served for two years as chair of WIMSIG, the Women in Mathematics group of the Australian Mathematical Society. She is the Associate Dean of Research in Monash's Faculty of Science, and President-elect of the Australian Mathematical Society. She strives to make mathematical research more inclusive.

Financial Support

Financial support for the workshop series is provided by a grant from the National Science Foundation (DMS-1764311).

Organizers

Fredric Ancel, University of Wisconsin-Milwaukee
Greg Friedman, Texas Christian University
Craig Guilbault, University of Wisconsin-Milwaukee
Molly Moran, Colorado College
Nathan Sunukjian, Calvin College
Eric Swenson, Brigham Young University
Frederick Tinsley, Colorado College
Gerard Venema, Calvin College

Please contact Greg Friedman (send email) if you have questions about the workshop or comments on this web site.