Columbia Geometric Topology Seminar

Spring 2020


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.

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Fall 2019

Date Speaker Title
January 24 Ryan Budney Isotopy in dimension 4
January 31 Nick Salter Framed mapping class groups and strata of abelian differentials
February 7 Jacob Russell Relative hyperbolicity in hierarchically hyperbolic spaces
February 14 Colin Adams Hyperbolicity and Turaev Hyperbolicity of Classical and Virtual Knots
February 21 Daniele Alessandrini Non commutative cluster coordinates for Higher Teichmüller Spaces
February 28 Abhijit Champanerkar Right-angled polyhedra and alternating links 
March 6 Peter Shalen Quantitative Mostow Rigidity
March 13 Lei Chen Actions of Homeo and Diffeo on manifolds
March 27 Keiko Kawamuro  
April 3 Talia Fernos  
April 10 Harrison Bray  
April 17    
April 24 Rylee Lyman  
May 1 Hans Boden  
May 8    
May 15 Johanna Mangahas  

Abstracts

Lei Chen, Caltech

March 13
Title: Actions of Homeo and Diffeo on manifolds
Abstract
: In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is a joint work with Kathryn Mann.

Abhijit Champanerkar, CUNY

February 28
Title
Right-angled polyhedra and alternating links 

Abstract: To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link.  We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link. This is joing work with Ilya Kofman and Jessica Purcell. 

Daniele Alessandrini, Columbia

February 21
Title
:Non commutative cluster coordinates for Higher Teichmüller Spaces

Abstract: In higher Teichmuller theory we study subsets of the character varieties

of surface groups that are higher rank analogs of Teichmuller spaces,
e.g. the Hitchin components and the spaces of maximal representations.

Fock-Goncharov generalized Thurston's shear coordinates and Penner's
Lambda-lengths to the Hitchin components, showing that they have a
beautiful structure of cluster variety.

Here we apply similar ideas to Maximal Representations and we find new
coordinates on these spaces that give them a structure of non-commutative
cluster varieties, in the sense defined by Berenstein-Rethak.

This is joint work with Guichard, Rogozinnikov and Wienhard.

 

Colin Adams, Williams College

February 14
Title
Hyperbolicity and Turaev Hyperbolicity of Classical and Virtual Knots

Abstract: We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume.   We will discuss what is known about  this invariant. We further employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction.  We will talk about what is known.

 

Jacob Russell, CUNY

February 7
Title: Relative hyperbolicity in hierarchically hyperbolic spaces
Abstract
Relative hyperbolicity and thickness describe incompatible ways that the non-negatively curved parts of a metric space can be organized. In several classes of spaces (Teichmuller space, Coxeter groups, 3-manifold groups) there exists a strict dichotomy between relative hyperbolicity and thickness that  produces strong geometric consequences. Behrstock, Drutu, and Mosher have thus asked for which additional classes of spaces can such a dichotomy be established. We investigate this question in the class of hierarchically hyperbolic spaces and produce a combinatorial criteria for detecting relative hyperbolicity. We apply this criteria to prove the separating curve graph of a surface has the relatively hyperbolic versus thick dichotomy.

 

Nick Salter, Columbia University

January 31
Title: Framed mapping class groups and strata of abelian differentials
Abstract: A holomorphic 1-form on a Riemann surface admits a geometric incarnation as a so-called translation surface. The moduli spaces of translation surfaces are known as strata. The dynamics of translation surfaces is an intense area of active study, but the topological properties of strata are almost entirely unknown. I will outline some work, joint with Aaron Calderon, aimed at obtaining information about the fundamental groups of strata by means of a monodromy representation into the mapping class group. The core of our approach is a study of the ``framed mapping class group'', a natural infinite-index subgroup that, very surprisingly, turns out to admit a very simple finite set of generators.

 

Ryan Budney, University of Victoria

January 24
Title: Isotopy in dimension 4
Abstract
I will describe why the trivial knot S2-->S4 has non-unique spanning discs up to isotopy. This comes from a chain of deductions that include a description of the low-dimensional homotopy-groups of embeddings of S1 in S1xSn (for n>2), a group structure on the isotopy-classes of reducing discs of S1xDn, and the action of the diffeomorphism group Diff(S1xSn) on the embedding space Emb(S1, S1xSn).  Roughly speaking, these results say there is no direct translation from dimension 3 to 4, for the Hatcher-Ivanov theorems on spaces of incompressible surfaces. 

Other relevant information.

Previous semesters:

Fall 2019Spring 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.

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