Codes and Expansions (CodEx) Seminar

Samuel Ballas (Florida State University):
Frame theory for vector bundles

Abstract: Say you want to design a system to describe wind velocities at each point on the surface of the earth. The surface of the earth is a 2-dimensional manifold, and so you might imagine that you will only need 2 parameters to describe the vectors at each point. However, here the topology of the sphere gets in the way and you will eventually find that there is no way to globally describe wind velocities using 2 parameters. However, if you embed the sphere in 3-dimensional space then you can use the 3 coordinates there to globally parameterize wind velocities. The situation above is a manifestation of a general problem: given a vector bundle, it is often impossible to choose a global basis, however it may nonetheless be possible to choose a global frame. In this talk I will discuss some recent work with T. Needham and C. Shonkwiler where we show that it is always possible to find a frame on a vector bundle over smooth manifold and that the size of the frame grows “linearly” with the size of the vector bundle.