Softcover ISBN:  9781470469610 
Product Code:  GSM/220.S 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Electronic ISBN:  9781470469603 
Product Code:  GSM/220.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 

Book DetailsGraduate Studies in MathematicsVolume: 220; 2022; 399 ppMSC: Primary 03; 54;
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.ReadershipUndergraduate and graduate students and researchers interested in ultrafilters and ultraproducts in geometric group theory, combinatorics, and number theory.

Table of Contents

Ultrafilters and their applications

Ultrafilter basics

Arrow’s theorem on fair voting

Ultrafilters in topology

Ramsey theory and combinatorial number theory

Foundational concerns

Classical ultraproducts

Classical ultraproducts

Applicationis to geometry, commutative algebra, and number theory

Ultraproducts and saturation

Nonstandard analysis

Limit groups

Metric ultraproducts and their applications

Metric ultraproducts

Asymptotic cones and Gromov’s theorem

Sofic groups

Functional analysis

Advanced topics

Does an ultrapower depend on the ultrafilter?

The KeislerShelah theorem

Large cardinals

Appendices

Logic

Set theory

Category theory

Hints and solutions to selected exercises


Additional Material

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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.
Undergraduate and graduate students and researchers interested in ultrafilters and ultraproducts in geometric group theory, combinatorics, and number theory.

Ultrafilters and their applications

Ultrafilter basics

Arrow’s theorem on fair voting

Ultrafilters in topology

Ramsey theory and combinatorial number theory

Foundational concerns

Classical ultraproducts

Classical ultraproducts

Applicationis to geometry, commutative algebra, and number theory

Ultraproducts and saturation

Nonstandard analysis

Limit groups

Metric ultraproducts and their applications

Metric ultraproducts

Asymptotic cones and Gromov’s theorem

Sofic groups

Functional analysis

Advanced topics

Does an ultrapower depend on the ultrafilter?

The KeislerShelah theorem

Large cardinals

Appendices

Logic

Set theory

Category theory

Hints and solutions to selected exercises