Geometry and Topology Seminar at Brown and Yale (GATSBY)

November 18, 2023 at Brown University

GATSBY is a conference organized by the geometry/topology groups at Brown and Yale. 

All are welcome! 

Speakers

Jeffrey Danciger (UT Austin)

Schedule (talks in MacMillan 115)

10-10:30 Coffee, pastries, fruit

10:30-11:15 Danciger background talk

11:15-11:45 Break

11:45-12:30 Frankel background talk

12:30-2 Lunch (catered)

2-2:45 Danciger research talk

2:45-3:15 Break

3:15-4 Frankel research talk

Abstracts



Background talk: Thurston’s asymmetric metric on Teichmüller space revisited


We recast the story of Thurston’s asymmetric metric on Teichmüller space as a story about the Benoist limit cones of surface group representations into the rank two semi-simple Lie group PSL_2 R ⨉ PSL_2 R. We give a new argument explaining why the supremum of length ratios over all homotopy classes of curves is attained over the subset of simple curves. 


Research talk: Record breakers, slack calculus, and the Benoist limit cone


We study the boundary of the Benoist limit cone of a positive representation from a surface group into a semi-simple Lie group G. To keep the analogy with the first talk, we focus mainly on the case G = (PSL_2 R)^n. Here the Benoist limit cone is obtained by plotting in R^n the n-tuple of translation lengths of each element of the group, and then taking the closure of the cone spanning these points. It turns out to be convex. We investigate the question of which elements of the group (and more generally, which geodesic currents) appear on the boundary. Joint work with François Guéritaud and Fanny Kassel.




Background talk: Quasigeodesic and pseudo-Anosov flows:

 

Quasigeodesic and pseudo-Anosov flows are defined by simple geometric conditions, and interact in surprising ways with the geometry of 3-manifolds. We will introduce these flows by illustrating their origins, in Cannon and Thurston’s construction of sphere-filling curves from hyperbolic surface bundles.

 

Research talk: Flowspaces, universal circles, and orbit equivalences


The transverse structure of a quasigeodesic or pseudo-Anosov flow is captured by its 2-dimensional flowspace, a topological plane equipped with an action of the ambient manifold’s fundamental group. This is compactified and simplified by a 1-dimensional universal circle. We will discuss the surprising extent to which these lower-dimensional discrete systems reflect the dynamics of their associated flows. In particular, we will discuss joint work with Thomas Barthelme and Kathryn Mann showing that for many pseudo-Anosov flows, the set of free homotopy classes of closed orbits determines the flowspace up to conjugacy, and this in turn determines the flow up to orbit equivalence.


Local information


The math department is in "Kassar House" on 151 Thayer Street, at the corner of George and Thayer Streets. The talks are in MacMillan, which is opposite of Kassar on George Street.  


Parking: There is visitor parking at 111 Power Street (~$15-20/day). You can also find free street parking on Cooke Street (e.g. in front of the Rhode Island historical society -- see pin on map below), or in that general area. If you look for something else, check that there are no time-restricted parking signs on the street where you are parking. 



Previous iterations: Spring 2023, Fall 2022, Spring 2019, Fall 2015, Spring  2015, Fall 2014, Spring 2014, Fall 2013,  Spring 2013, Spring 2012