The representation theory of Kac-Moody algebras. There are some issues with integration theories that need clearing up, should we really be ditching the Riemann integral for the Lebesgue integral so quickly without a closer look. Recently someone wrote a good book on linear algebra that puts of "ugly" determinants for as long as possible. Much of the problem is making what we already know cleaner and better. Some theories like set theory don't have a unique set of axioms and have lots of loose ends. General topology is a closed subject but there are still potentially better ways to formulate convergence.