EVENT DETAILS AND ABSTRACT

Title: Immersions of surfaces and the Arf invariant
Speaker: Mike Hopkins
Speaker Info: Harvard University
Brief Description:
Special Note: ANNUAL UNDERGRADUATE PRIZE LECTURE
Abstract:

A surface is any space that looks locally like the plane. The sphere is an example we use all the time: we project some small regions of the globe onto the flat planes we call maps. However, as the example of the torus or the Klein bottle shows, surfaces can have very interesting geometry, so interesting in fact, that we sometimes can't draw them in 3-space without overlaps or, put another way, we can only immerse them in 3-space. Can we list all immersed surfaces? That is, can we give a classification? The story involves a mix of deep ideas from algebra and geometry, among them Cahit Arf's famous invariant (which is honored on the 2009 Turkish 10 Lire note). I will describe the classification, Arf's invariant and, time permitting, some of the ways these ideas occur in other problems.