Arthur H. Vartanian Department of Mathematics, College of Charleston, U.S.A. Orthogonal rational functions Wednesday 14th June 14:05-14:55pm, Carslaw Building Room 373. Orthogonal Rational Functions (ORFs) are rational functions with no poles in the extended complex plane outside of the real pole set {alpha_1,...,alpha_K}, with 0\leq |alpha_k|<+\infty, k=1,...,K, which satisfy a system of orthogonality conditions; in particular, ORFs orthogonal with respect to varying exponential weights \varpi(z)=\exp(-N V(z)), N\in\mathbb{N}, where the external field, V, is real analytic and satisfies certain `growth conditions' at \infty and at alpha_k, k=1,...,K, are considered. The principal question addressed in this talk is: what are asymptotics in a double-scaling limit as $\mathscr{N}$ and n, the `degree' of the ORF, tend to infinity in such a way that $\mathscr{N}/n=1+o(1)$ for z anywhere in the extended complex plane and each k=1,...,K? This will be addressed by considering the Fokas-Its-Kitaev reformulation of the ORF problem as a matrix Riemann-Hilbert problem, and then extracting the asymptotic behaviour by applying the Deift-Zhou non-linear steepest-descent method. Possible applications of ORFs to Multi-Point Pade' Approximants, Random Matrix Theory, and Painleve' Transcendents will be discussed.