Title: p-points and their opposites

Abstract:

An ultrafilter U on omega (the natural numbers) is a p-point if
for every family (A_n: n in omega) of elements of U there is a
pseudointersection B in U, i.e. a set B which is almost contained
in every A_n ( B minus A_n is finite).
It is known that CH implies that p-points exist, and that
"there are no p-points" is consistent with ZFC.

If U is an ultrafilter on X, V an ultrafilter on Y,
we say that V is a Rudin-Keisler quotient of U if there is
a function f: X -> Y such that  V = { B:   f^-1[B] in U }.

I will review these concepts,  explain the p-point game,
and sketch a construction of Shelah that yields an ultrafilter
without any Rudin-Keisler p-point quotient.  I will also
talk briefly about work in progress:  the construction of
a set-theoretic universe in which no p-points exist, yet
there is an ultrafilter generated by aleph1 many sets.