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Ultrafilters Throughout Mathematics

Isaac Goldbring University of California, Irvine, Irvine, CA
Available Formats:
Softcover ISBN: 978-1-4704-6961-0
Product Code: GSM/220.S
List Price: $85.00 MAA Member Price:$76.50
AMS Member Price: $68.00 Electronic ISBN: 978-1-4704-6960-3 Product Code: GSM/220.E List Price:$85.00
MAA Member Price: $76.50 AMS Member Price:$68.00
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $127.50 MAA Member Price:$114.75
AMS Member Price: $102.00 Click above image for expanded view Ultrafilters Throughout Mathematics Isaac Goldbring University of California, Irvine, Irvine, CA Available Formats:  Softcover ISBN: 978-1-4704-6961-0 Product Code: GSM/220.S  List Price:$85.00 MAA Member Price: $76.50 AMS Member Price:$68.00
 Electronic ISBN: 978-1-4704-6960-3 Product Code: GSM/220.E
 List Price: $85.00 MAA Member Price:$76.50 AMS Member Price: $68.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$127.50 MAA Member Price: $114.75 AMS Member Price:$102.00
• Book Details

Graduate Studies in Mathematics
Volume: 2202022; 399 pp
MSC: 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.

Readership

Undergraduate 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 Keisler-Shelah theorem
• Large cardinals
• Appendices
• Logic
• Set theory
• Category theory
• Hints and solutions to selected exercises
• Additional Material

• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2202022; 399 pp
MSC: 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.

Readership

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 Keisler-Shelah theorem
• Large cardinals
• Appendices
• Logic
• Set theory
• Category theory
• Hints and solutions to selected exercises
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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