Department of Mathematics

University of Chicago

**Abstract**: In this course we will discuss the method of Carleman
estimates to prove uniqueness theorems. We will start out by recalling
Carleman's
classical unique continuation theorem for second order elliptic equations
and then obtaining some new quantitative versions of it. These are
instrumental in the recent proof by Bourgain-Kenig of Anderson
localization for the Bernoulli model in higher dimensions. We will then
move on to analogs for parabolic operators (joint work of Escauriaza,
Kenig, Ponce and Vega) which settles a conjecture of Landis-Oleinik. This
is in turn connected to the work of Escauriaza-Seregin-Sverak (2002) on
regularity of a class of weak solutions to Navier-Stokes. Finally, we will
discuss versions of these quantitative uniqueness results for dispersive
equations, such as non-linear Schrodinger and KdV equations. (This is
also joint work of Escauriaza. Kenig, Ponce and Vega.) These results
provide a natural extension of Hardy's uncertainty principle to dispersive
equations.