"Negative Dependence and the Symmetric Exclusion Process" Thomas Liggett, UCLA Over the past several years, several conjectures related to negative dependence of Bernoulli random variables have been made. Among them are: (a) the Rayleigh property (also known as the hereditary negative lattice condition after application of external fields) implies the ultra logconcavity (ULC) of the rank sequence, (b) the Rayleigh property implies negative association, and (c) the symmetric exclusion process with product initial distribution is negatively associated at positive times. We will discuss these and other conjectures. Among the results: (a) is false and (c) is true, while (b) is still open. Furthermore, a stronger form of the Rayleigh property does imply both ULC and negative association. As a consequence of (c), we obtain distributional limit theorems for certain functionals of the symmetric exclusion process. Much of this is joint work with J. Borcea and P. Branden.