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Geometry-Topology Seminar

Friday, April 22, 2022 - 9:30am

Hannah Schwartz and Franco Vargas Pallete

Princeton and Yale


University of Pennsylvania


This is a full day meeting, sponsored jointly with Bryn Mawr, Haverford and Temple. The talks will be given in person in DRL A2, and will also be streamed live on Zoom. Please e-mail for the Zoom link as needed.

9:30 AM in DRL A2   Hannah Schwartz  "Pretty pictures of regular homotopies"

This introductory talk will be largely picture-based, and will offer geometric interpretations of the Freedman-Quinn and Dax invariants relevant to the afternoon lecture. In particular, we will showcase examples in which each does not vanish and so obstructs isotopy between pairs of homotopic surfaces.

11 AM in DRL A2   Franco Vargas Pallete   "Volume in Hyperbolic Geometry"

We will see the role that volume plays in hyperbolic geometry. We will prove a version of volume rigidity and introduce the Bonahon-Schlafli formula. Both statements will prove extremely useful for the second part of this talk.

2 PM in DRL A2   Hannah Schwartz   "Isotopy vs. homotopy for disks with a common dual"

Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90's, along with homotopy theoretic isotopy invariants of Dax from the 70's. We will outline, give context to, and discuss techniques used to prove these so called "light bulb theorems", and present new light bulb theorems for disks rather than spheres.   

3:30 PM in DRL A2   Franco Vargas Pallete "Peripheral birationality for 3-dimensional convex co-compact  PSL(2, C)  varieties"

It is a consequence of a well-known result of Ahlfors and Bers that the PSL(2, C) character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding PSL(2, C) character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.