Not your grandparents' linear algebra

Here's a fun brainteaser: suppose I give you a collection of n x n matrices. Can you tell whether some linear combination of them is an invertible matrix? If you can, then you probably have a Fields medal coming your way—this would be the most significant progress on the P vs. NP problem in over 50 years.

Questions of this type lie in a surprisingly rich area of math, where one combines the two most basic notions in linear algebra—vector spaces and matrices—by studying vector spaces whose elements are matrices. In this talk, I'll discuss some of these questions and what is known about them, stressing their connections to a shockingly wide array of topics, including algebraic geometry, graph theory, theoretical computer science, and topological K-theory.