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.
Author(s): Isaac Goldbring
Series: Graduate Studies in Mathematics, 220
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 420
City: Providence
Cover
Title page
Preface
Part 1. Ultrafilters and their applications
Chapter 1. Ultrafilter basics
1.1. Basic definitions
1.2. The ultrafilter quantifier
1.3. The category of ultrafilters
1.4. The number of ultrafilters
1.5. The ultrafilter number ?
1.6. The Rudin-Keisler order
1.7. Notes and references
Chapter 2. Arrow’s theorem on fair voting
2.1. Statement of the theorem
2.2. The connection with ultrafilters
2.3. Block voting
2.4. Finishing the proof
2.5. Notes and references
Chapter 3. Ultrafilters in topology
3.1. Ultralimits
3.2. The Stone-Čech compactification: the discrete case
3.3. ?-ultrafilters and the Stone-Čech compactifications in general
3.4. The Stone representation theorem
3.5. Notes and References
Chapter 4. Ramsey theory and combinatorial number theory
4.1. Ramsey’s theorem
4.2. Idempotent ultrafilters and Hindman’s theorem
4.3. Banach density, means, and measures
4.4. Furstenberg’s correspondence principle
4.5. Jin’s sumset theorem
4.6. Notes and references
Chapter 5. Foundational concerns
5.1. The ultrafilter theorem and the axiom of choice: Part I
5.2. Can there exist a “definable” ultrafilter on ℕ?
5.3. The ultrafilter game
5.4. Selective ultrafilters and P-points
5.5. Notes and references
Part 2. Classical ultraproducts
Chapter 6. Classical ultraproducts
6.1. Motivating the definition of ultraproducts
6.2. Ultraproducts of sets
6.3. Ultraproducts of structures
6.4. Łoś’s theorem
6.5. The ultrafilter theorem and the axiom of choice: Part II
6.6. Countably incomplete ultrafilters
6.7. Revisiting the Rudin-Keisler order
6.8. Cardinalities of ultraproducts
6.9. Iterated ultrapowers
6.10. A category-theoretic perspective on ultraproducts
6.11. The Feferman-Vaught theorem
6.12. Notes and references
Chapter 7. Applications to geometry, commutative algebra, and number theory
7.1. Ax’s theorem on polynomial functions
7.2. Bounds in the theory of polynomial rings
7.3. The Ax-Kochen theorem and Artin’s conjecture
7.4. Notes and references
Chapter 8. Ultraproducts and saturation
8.1. Saturation
8.2. First saturation properties of ultraproducts
8.3. Regular ultrafilters
8.4. Good ultrafilters: Part 1
8.5. Good ultrafilters: Part 2
8.6. Keisler’s order
8.7. Notes and references
Chapter 9. Nonstandard analysis
9.1. Naïve axioms for nonstandard analysis
9.2. Nonstandard numbers big and small
9.3. Some nonstandard calculus
9.4. Ultrapowers as a model of nonstandard analysis
9.5. Complete extensions and limit ultrapowers
9.6. Many-sorted structures and internal sets
9.7. Nonstandard generators of ultrafilters
9.8. Hausdorff ultrafilters
9.9. Notes and references
Chapter 10. Limit groups
10.1. Introducing the class of limit groups
10.2. First examples and properties of limit groups
10.3. Connection with fully residual freeness
10.4. Explaining the terminology: the space of marked groups
10.5. Notes and references
Part 3. Metric ultraproducts and their applications
Chapter 11. Metric ultraproducts
11.1. Definition of the metric ultraproduct
11.2. Metric ultraproducts and nonstandard hulls of metric spaces
11.3. Completeness properties of the metric ultraproduct
11.4. Continuous logic
11.5. Reduced products of metric structures
11.6. Notes and references
Chapter 12. Asymptotic cones and Gromov’s theorem
12.1. Some group-theoretic preliminaries
12.2. Growth rates of groups
12.3. Gromov’s theorem on polynomial growth
12.4. Definition of asymptotic cones
12.5. General properties of asymptotic cones
12.6. Growth functions and properness of the asymptotic cones
12.7. Properness of asymptotic cones revisited
12.8. Nonhomeomorphic asymptotic cones
12.9. Notes and references
Chapter 13. Sofic groups
13.1. Ultraproducts of bi-invariant metric groups
13.2. Definition of sofic groups
13.3. Examples of sofic groups
13.4. An application of sofic groups
13.5. Notes and references
Chapter 14. Functional analysis
14.1. Banach space ultraproducts
14.2. Applications to local geometry of Banach spaces
14.3. Commutative ?*-algebras and ultracoproducts of compact spaces
14.4. The tracial ultraproduct construction
14.5. The Connes embedding problem
14.6. Notes and references
Part 4. Advanced topics
Chapter 15. Does an ultrapower depend on the ultrafilter?
15.1. Statement of results
15.2. The case when ℳ is unstable
15.3. The case when ℳ is stable
15.4. Notes and references
Chapter 16. The Keisler-Shelah theorem
16.1. The Keisler-Shelah theorem
16.2. Application: Elementary classes
16.3. Application: Robinson’s joint consistency theorem
16.4. Application: Elementary equivalence of matrix rings
16.5. Notes and references
Chapter 17. Large cardinals
17.1. Worldly cardinals
17.2. Inaccessible cardinals
17.3. Measurable cardinals
17.4. Strongly and weakly compact cardinals
17.5. Ramsey cardinals
17.6. Measurable cardinals as critical points of elementary embeddings
17.7. An application of large cardinals
17.8. Notes and references
Part 5. Appendices
Appendix A. Logic
A.1. Languages and structures
A.2. Syntax and semantics
A.3. Embeddings
A.4. References
Appendix B. Set theory
B.1. The axioms of ???
B.2. Ordinals
B.3. Cardinals
B.4. ? and ?
B.5. Relative consistency statements
B.6. Relativization and absoluteness
B.7. References
Appendix C. Category theory
C.1. Categories
C.2. Functors, natural transformations, and equivalences of categories
C.3. Limits
C.4. References
Appendix D. Hints and solutions to selected exercises
Bibliography
Index
Back Cover
