Kathryn Mann (Cornell)

Math & Stats Department Colloquium
Friday, September 20, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Kathryn Mann (Cornell)

Title: From the plane to infinity and back again

Abstract: A "bifoliation" of a two-dimensional space is a way of covering it with local charts to the Euclidean plane $\mathbb R^2$ so that overlap maps in $\mathbb R^2$ match up the vertical and horizontal coordinate directions. Such objects arise naturally in many dynamical contexts such as Anosov diffeomorphisms on surfaces or flows on 3-manifolds.

A trick due to Mather lets one compactify a bifoliated plane with a "circle at infinity" using the data of the bifoliation. In recent work with Barthelm\'e and Bonatti, we studied the inverse question: what is the minimum amount of data from infinity that allows one to reverse this procedure and uniquely reconstruct a bifoliation of the plane?

This talk will introduce bifoliations and where they come from, and then answer the question above. While we are motivated by problems in dynamics, the talk will be mostly delightfully low-tech, low-dimensional topology, with lots of believable pictures.