Math 295-22 (Salle Morrissey)
Description:
Ultrafilters on a set X are maximal (under inclusion) subsets of the power
set of X
which are upward closed, closed under finite intersection, and do
not contain the empty set. From
this simple definition arises a very rich
set of ideas,
developed over the course of the twentieth century.
Ultrafilters over infinite sets
give a generalized notion of limit, and they
also allow us to compute
averages of infinitely many
objects via the
ultraproduct construction. We will cover the basics of
ultrafilters and
ultraproducts
and the building of different types of ultrafilters, and
discuss some
applications in combinatorics,
logic, topology and algebra as
time permits. The course will have
a final project.
No special prerequisites
beyond the program requirements will be assumed.
Sources: Class notes and handouts.
Suggested further reading: Chang and Keisler, Model Theory, third edition.