Mathematical Conversations

Classical probability theory is set up to handle random variables whose values are a single complex number. What happens if our random variable is instead a multi-set of complex numbers? For example, on the group of n by n orthogonal matrices, you might want to study the random variable that sends a matrix to its eigenvalues. Or, if you're a number theorist, you might want to consider the random variable on the finite set of smooth plane curves over a finite field $F_q$ that sends a curve C to the zeroes of its L-function. In this conversation, we'll explain how a theory of probability for random variables valued in an algebraic structure called a lambda ring can help study these questions. As a concrete example, we will sketch how this perspective can be used to give a simple proof a theorem of Diaconis and Shahshahani on traces of powers in classical groups, and how this relates to the behavior of zeros of L-functions.