Ultrafilters and ultraproducts provide a useful generalization of
the ordinary limit processes which have applications to many areas
of mathematics. Typically, this topic is presented to students in
specialized courses such as logic, functional analysis, or
geometric group theory. In this book, the basic facts about
ultrafilters and ultraproducts are presented to readers with no
prior knowledge of the subject and then these techniques are
applied to a wide variety of topics. The first part of the book
deals solely with ultrafilters and presents applications to voting
theory, combinatorics, and topology, while also dealing also with
foundational issues. The second part presents the classical
ultraproduct construction and provides applications to algebra,
number theory, and nonstandard analysis. The third part discusses a
metric generalization of the ultraproduct construction and gives
example applications to geometric group theory and functional
analysis. The final section returns to more advanced topics of a
more foundational nature. The book should be of interest to
undergraduates, graduate students, and researchers from all areas
of mathematics interested in learning how ultrafilters and
ultraproducts can be applied to their specialty.
General
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