Chris Ormerod University of Sydney Associated linear problems for ultradiscrete equations Wednesday 18th April 14:05-14:55pm, Eastern Avenue Lecture Theatre. One may consider the appropriate domain for ultradiscrete equations to be the max-plus semiring. Using techniques from tropical algebraic geometry, we develop the theory of Birkhoff for systems of linear difference equations over the max-plus semiring. We use such theory to provide evidence for the integrability of some ultradiscrete difference equations. We show that by considering Birkhoffs fundamental solutions, we may recover the first known hypergeometric solutions for the ultradiscrete Painlev\'e equations.