Thomas Barthelmé (Queen's University)

Math Club

Thursday, February 8th, 2024

Time: 5:30 p.m.  Place: Jeffery Hall, Room 118

Speaker: Thomas Barthelmé (Queen’s University)

Title: Bifoliations of the plane to prelaminations of the circle and back again

Abstract: Think of the plane as an (infinite) plate. Then a foliation of the plane is like (infinite) spaghettis on your plate but arranged so that every point of the plate is covered by one and only one spaghetto.This is joint work with Kathyrn Mann and Christian Bonatti.

An old (1940s) result of Kaplan, improved by Mather (1982), is that given your plate of spaghettis, you can make a border of your plate so that each spaghetto is attached at exactly two points of the border of the plate.This is joint work with Kathyrn Mann and Christian Bonatti.

Now a bifoliation of the plane is a plate with two types of spaghetti (some reds and some greens) such that through every point of the plate passes two spaghetti (one red and one green) and such that all the red spaghettis are transverse (ie crosses) the green spaghettis. Similarly, one can add a border to the plate so that each spaghetto ends at exactly two distinct points of the boundary (in that generality, this result is due to Bonatti, in 2023). This set of ends is an example of a prelamination of the circle.This is joint work with Kathyrn Mann and Christian Bonatti.

After describing more precisely the results above, and how one can obtain them, I'll discuss the opposite direction: Given a set of pairs of points on the border of the plate, what are sufficient conditions so that these are the ends of a platter of green/red spaghettis?This is joint work with Kathyrn Mann and Christian Bonatti.

This is joint work with Kathyrn Mann and Christian Bonatti.