Introduction to Total Positivity

Lauren Williams, UC Berkeley

The classical theory of total positivity is the study of real n by n matrices such that all minors are positive (or nonnegative). Such matrices appear in various areas of mathematics such as graph theory, algebraic geometry, stochastic processes, game theory, matroids, and representation theory. They also appear in various areas of study in the natural sciences, including brownian motion and electrical networks. In this talk we will survey some of the classical results about total positivity. Additionally, we will present some recent work on totally nonnegative real flag varieties.