Ultrafilters throughout mathematics / Isaac Goldbring.

By:

Material type: Text

TextLanguage: English Series: ; v.220 | Graduate studies in mathematics ; 220Description: xviii, 399 pages ; 26 cmISBN:

9781470469009
9781470469610

Subject(s):

DDC classification:

511.3 23/eng/20220112 GOL

Other classification:

03C20 | 54D80 | 03H05 | 03E55 | 03C50

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

Summary: Graduate Studies in Mathematics Volume: 220; 2022; 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.

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Holdings
Item type	Current library	Collection	Call number	Status	Date due	Barcode
Project book	CUTN Central Library Sciences	Non-fiction	511.3 GOL (Browse shelf(Opens below))	Checked out to Renuka Devi V (20019T)	31/01/2024	48925

Includes bibliographical references (pages 385-394) and index.

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

Graduate Studies in Mathematics
Volume: 220; 2022; 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.

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