**Graduate Studies in Mathematics**

Volume: 220;
2022;
399 pp;
Softcover

MSC: Primary 03; 54;

**Print ISBN: 978-1-4704-6961-0
Product Code: GSM/220.S**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

**Electronic ISBN: 978-1-4704-6960-3
Product Code: GSM/220.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

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#### Supplemental Materials

# Ultrafilters Throughout Mathematics

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*Isaac Goldbring*

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

# Table of Contents

## Ultrafilters Throughout Mathematics

- Cover Cover11
- Title page iii4
- Preface xiii14
- Part 1. Ultrafilters and their applications 120
- Part 2. Classical ultraproducts 7998
- Chapter 6. Classical ultraproducts 81100
- 6.1. Motivating the definition of ultraproducts 82101
- 6.2. Ultraproducts of sets 83102
- 6.3. Ultraproducts of structures 85104
- 6.4. Łoś’s theorem 86105
- 6.5. The ultrafilter theorem and the axiom of choice: Part II 89108
- 6.6. Countably incomplete ultrafilters 92111
- 6.7. Revisiting the Rudin-Keisler order 94113
- 6.8. Cardinalities of ultraproducts 96115
- 6.9. Iterated ultrapowers 98117
- 6.10. A category-theoretic perspective on ultraproducts 101120
- 6.11. The Feferman-Vaught theorem 104123
- 6.12. Notes and references 108127

- Chapter 7. Applications to geometry, commutative algebra, and number theory 109128
- Chapter 8. Ultraproducts and saturation 123142
- Chapter 9. Nonstandard analysis 157176
- 9.1. Naïve axioms for nonstandard analysis 157176
- 9.2. Nonstandard numbers big and small 159178
- 9.3. Some nonstandard calculus 161180
- 9.4. Ultrapowers as a model of nonstandard analysis 163182
- 9.5. Complete extensions and limit ultrapowers 164183
- 9.6. Many-sorted structures and internal sets 168187
- 9.7. Nonstandard generators of ultrafilters 173192
- 9.8. Hausdorff ultrafilters 177196
- 9.9. Notes and references 178197

- Chapter 10. Limit groups 181200

- Part 3. Metric ultraproducts and their applications 193212
- Chapter 11. Metric ultraproducts 195214
- 11.1. Definition of the metric ultraproduct 195214
- 11.2. Metric ultraproducts and nonstandard hulls of metric spaces 198217
- 11.3. Completeness properties of the metric ultraproduct 199218
- 11.4. Continuous logic 201220
- 11.5. Reduced products of metric structures 208227
- 11.6. Notes and references 209228

- Chapter 12. Asymptotic cones and Gromov’s theorem 211230
- 12.1. Some group-theoretic preliminaries 212231
- 12.2. Growth rates of groups 213232
- 12.3. Gromov’s theorem on polynomial growth 216235
- 12.4. Definition of asymptotic cones 221240
- 12.5. General properties of asymptotic cones 223242
- 12.6. Growth functions and properness of the asymptotic cones 226245
- 12.7. Properness of asymptotic cones revisited 229248
- 12.8. Nonhomeomorphic asymptotic cones 231250
- 12.9. Notes and references 232251

- Chapter 13. Sofic groups 233252
- Chapter 14. Functional analysis 249268
- 14.1. Banach space ultraproducts 249268
- 14.2. Applications to local geometry of Banach spaces 254273
- 14.3. Commutative 𝐶*-algebras and ultracoproducts of compact spaces 259278
- 14.4. The tracial ultraproduct construction 264283
- 14.5. The Connes embedding problem 273292
- 14.6. Notes and references 277296

- Part 4. Advanced topics 279298
- Chapter 15. Does an ultrapower depend on the ultrafilter? 281300
- Chapter 16. The Keisler-Shelah theorem 297316
- Chapter 17. Large cardinals 309328
- 17.1. Worldly cardinals 309328
- 17.2. Inaccessible cardinals 311330
- 17.3. Measurable cardinals 314333
- 17.4. Strongly and weakly compact cardinals 320339
- 17.5. Ramsey cardinals 325344
- 17.6. Measurable cardinals as critical points of elementary embeddings 327346
- 17.7. An application of large cardinals 333352
- 17.8. Notes and references 338357

- Part 5. Appendices 339358
- Back Cover Back Cover1421