Title: On Global-in-Time Strichartz Estimates for the Semiperiodic Schrödinger Equation

Abstract: We will discuss some recent results related to space-time

estimates for solutions to the linear Schrödinger equation on

manifolds which are products of tori and Euclidean space (e.g. a

cylinder embedded in R^3). On these manifolds it is possible to prove

certain analogues of the classical Euclidean Strichartz estimates

which are scale-invariant and global-in-time. These estimates -- which

are proved with harmonic analysis techniques -- are strong enough to

imply small-data scattering for solutions to the critical quintic NLS

on RxT and the critical cubic NLS on R^2xT (where T is the

one-dimensional torus).