The asymmetric simple exclusion process (ASEP) is a Markov chain describing particles hopping on a 1-dimensional finite lattice.
The Waring problem asks whether for each positive integer n, every positive integer is the sum of a bounded number of nth powers. Although Hilbert solved it more than 100 years ago, it remains the focus of much current research.
There are several questions about the behavior of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved.
University of North Carolina at Chapel Hill
Ohio State University