Philip A. Treharne School of Mathematics and Statistics, University of Sydney The generalized Dirichlet to Neumann map for the KdV equation on the half-line Wednesday 9th August 14:05-14:55pm, Carslaw Building Room 373. For the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary conditition (if $q_{t}$ and $q_{xxx}$ have the same sign) or two boundary conditions (if $q_{t}$ and $q_{xxx}$ have opposite sign). In this talk I will construct the generalized Dirichlet to Neumann map for both versions of the KdV equation, i.e.~I will show how to characterise the unknown boundary values in terms of the given initial and boundary conditions. This construction involves analysis of the $t$-part of the associated Lax pair using a Gelfand--Levitan--Marchenko triangular representation and then consideration of a certain ``global relation'' which couples the given initial and boundary conditions with the unknown boundary values. This yields the unknown boundary values in terms of a nonlinear Volterra integral equation with an exponentially decaying kernel.