Abstract: I will show how recent connections between longest increasing subsequence problems and random matrices can be understood in the context of Pitman's theorem and its multidimensional extensions. Pitman's theorem gives a representation for a one-dimensional Brownian motion conditioned to stay positive as a certain functional of a (unconditioned) Brownian motion. Using ideas from queueing theory, this can be extended to a multidimensional setting, giving a representation for the eigenvalues of Hermitian Brownian motion and GUE random matrices (this is joint work with Marc Yor). As a corollary we recover an identity due to Baryshnikov and Gravner, Tracy and Widom, which relates a Brownian version of the longest increasing subsequence problem to the largest eigenvalue of a GUE random matrix. I will also indicate briefly how this is related to the RSK correspondence, and present some related results which can be obtained using this connection.