Ultrafilters Across Mathematics: International Congress Ultramath 2008: Applications of Ultrafilters and Ultraproducts in Mathematics, June 1-7, 2008, Pisa, Italy

214 Pages·2010·1.47 MB·English

by  Vitaly Bergelson

C
ONTEMPORARY
M
ATHEMATICS
530
Ultrafilters across Mathematics
International Congress
ULTRAMATH 2008: Applications of Ultrafilters
and Ultraproducts in Mathematics
June 1–7, 2008
Pisa, Italy
Vitaly Bergelson
Andreas Blass
Mauro Di Nasso
Renling Jin
Editors
American Mathematical Society
Ultrafilters across Mathematics
This page intentionally left blank
C
ONTEMPORARY
M
ATHEMATICS
530
Ultrafilters across Mathematics
International Congress
ULTRAMATH 2008: Applications of Ultrafilters
and Ultraproducts in Mathematics
June 1–7, 2008
Pisa, Italy
Vitaly Bergelson
Andreas Blass
Mauro Di Nasso
Renling Jin
Editors
American Mathematical Society
Providence, Rhode Island
Editorial Board
Dennis DeTurck, managing editor
George Andrews Abel Klein Martin J. Strauss
2000 Mathematics Subject Classification. Primary 03C20, 03E05,03H05, 05C55, 28E15.
Library of Congress Cataloging-in-Publication Data
ULTRAMATH2008: ApplicationsofUltrafiltersandUltraproductsinMathematics(2008: Pisa,
Italy)
Ultrafiltersacrossmathematics: internationalcongress,ULTRAMATH2008,Applicationsof
UltrafiltersandUltraproductsinMathematics,June1–7,2008,Pisa,Italy/VitalyBergelson...[et
al.],editors.
p.cm. —(Contemporarymathematics;v.530)
Includesbibliographicalreferences.
ISBN978-0-8218-4833-3(alk.paper)
1. Ultrafilters (Mathematics)—Congresses. 2. Combinatorial set theory—Congresses.
3.Ramseytheory—Congresses. I.Bergelson,V.(Vitaly),1950– II.Title.
QA9.7.I58 2010
511.3—dc22
2010025146
Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu-
cationaland scientific purposes without fee or permissionwith the exception ofreproduction by
servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg-
ment of the source is given. This consent does not extend to other kinds of copying for general
distribution, for advertising or promotional purposes, or for resale. Requests for permission for
commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath-
ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can
[email protected].
Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In
suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s).
(Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof
eacharticle.)
(cid:2)c 2010bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
Copyrightofindividualarticlesmayreverttothepublicdomain28years
afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 151413121110
Contents
Preface vii
An Elementary Approach to Stochastic Differential Equations Using the
Infinitesimals
Vieri Benci, Stefano Galatolo, and Marco Ghimenti 1
Ultrafilters, IP sets, Dynamics, and Combinatorial Number Theory
Vitaly Bergelson 23
Ultrafilters and Set Theory
Andreas Blass 49
Measure-Centering Ultrafilters
D. H. Fremlin 73
Algebra in the Space of Ultrafilters and Ramsey Theory
Neil Hindman and Dona Strauss 121
Ultrapower of N and Density Problems
Renling Jin 147
The Ultraproduct Construction
H. Jerome Keisler 163
Ultrafilters and Large Cardinals
Itay Neeman 181
v
This page intentionally left blank
Preface
The project of this volume originated from the International Conference:
ULTRAMATH2008: Applications ofUltrafilters andUltraproductsinMathematics
heldinPisa(Italy),June 1-7, 2008. ULTRAMATHaimedtopresent recentresults
inthewholespectrumofmathematicswhicharegroundedontheuseofultrafilters
and ultraproducts. One of its goals was to disseminate information about the
various ultrafilter techniques and their usefulness in dealing with open problems.
AnothergoaloftheULTRAMATHwastobringtogetherresearcherswithdifferent
backgrounds, and to facilitate their collaboration and interactions, especially on
topics which bridge different areas of mathematics. About 100 mathematicians
participated, and the program consisted of 12 invited talks and 32 contributed
talks (a complete list is given at the end of this preface).
Theintentionofthisvolumeistopresentthestate-of-the-artofapplicationsof
ultrafilter and ultraproduct techniques.
To this end, the editors chose experts on various aspects of ultrafilters, ultra-
powers, andtheirapplications, andinvitedthemtocontribute researchexpositions
orsurveysonthoseaspects. Allpapersincludedinthisvolumearethefinalversions
and they have all been individually refereed.
Besides two general surveys on ultrafilters in set theory and on the ultraprod-
uct construction, this volume includes papers that cover the following areas of
applications: additive and combinatorial number theory, nonstandard methods in
stochasticdifferentialequations,measuretheory,dynamics,Ramseytheory,algebra
in the space of ultrafilters, and large cardinals.
The papers in this volume are intended to be accessible and interesting for
students and researchers who are not experts on ultrafilters and ultraproducts.
Greaterprominencehasbeengiventoresultsthatcanbeformulatedandpresented
in non-specialist terms and be in principle understandable by any mathematician,
and to those results that connect different areas of mathematics, revealing new
facets of known important topics.
Acknowledgements. Wearegratefultothefollowingprogramsandorganizations
for the financial support that made “ULTRAMATH 2008” possible:
• Research program P.R.I.N. 2007 “Teoria dei Modelli, Teoria degli Insiemi e Ap-
plicazioni”;
vii
viii PREFACE
• Centro di Ricerca Matematica “E. De Giorgi”, Pisa;
• Istituto Nazionale di Alta Matematica I.N.D.A.M. - G.N.S.A.G.A.;
• School of Graduate Studies “Galileo Galilei”, Pisa;
• Universita` di Pisa;
• Journal “Logic and Analysis”, Springer.
WearealsogratefultotheA.M.S.foragreeingtopublishthevolume. Aspecial
thanks to Christine M. Thivierge, the American Mathematical Society’s Associate
Editor for Conference Proceedings, for her assistance and patience.
Vitaly Bergelson
Andreas Blass
Mauro Di Nasso
Renling Jin
PREFACE ix
ULTRAMATH 2008
Organizer: M. Di Nasso; Scientific committee: V. Bergelson, A. Blass, R. Jin;
WEB: http://www.dm.unipi.it/~ultramath.
List of invited talks. Vieri Benci, Nonstandard Methods in Analysis; Vitaly
Bergelson, Ultrafilters and Ergodic Ramsey Theory; AndreasBlass, Ultrafilters and
Set Theory; Nigel Cutland, Stochastic Navier-Stokes Equations: Ideas and Results
Using Nonstandard Analysis; David H. Fremlin, Ultrafilters in Measure Theory;
NeilHindman,AlgebraintheSpaceofUltrafiltersandRamseyTheory;RenlingJin,
Ultrapower of N and Density Problems; Marius Junge, Ultraproducts in Functional
Analysis; Melvyn B. Nathanson, Problems in Additive Number Theory with an
Ultrafilter Flavor; Itay Neeman, Ultrafilters, Determinacy, and Large Cardinals;
Jaroslav Neˇsetˇril, Some Topics on Non-Ramsey Connection of Combinatorics and
Ultrafilters; BobanVelickovic,Set Theoretic Aspects of the Space of Ultrafilters βN.
List of contributed talks. Mathias Beiglb¨ock, A Variant of the Hales-Jewett
Theorem;PiotrBorodulin-Nadzieja,OnPseudo-intersections andCondensers;Ab-
delmadjid Boudaoud, On the Terms of Unlimited Rank of Lucas Sequences; Ulf
Clotz,TopologyfromaRemotePointofView;DarylCooper,RealProjectiveStruc-
tures on Manifolds and the Hyper-reals; Abhijit Dasgupta, Logicless Nonstandard
Analysis: an Axiom System; Marco Forti, Counting Infinite Point Sets; Stefano
Francaviglia, Asymptotic Cones of Metric Spaces and Related Topics; Arthur D.
Grainger, Remarks on Ultrafilters on the Collection of Finite Subsets of an Infinite
Set; Gonsalo Gutierres, Ultrafilters, Closure Operators and the Axiom of Choice;
Frederik Herzberg, A Definable Framework for Nonstandard Analysis; Dirk Hof-
mann, Some Aspects of Ultrafilter Convergence in Topology; Karel Hrba`˘cek, Rela-
tive Set Theory; Peter Krautzberger, Idempotent Ultrafilters and Finer Topologies
on βN; Leonard Kwuida, Prime Ideal Theorem for Weakly Dicomplemented Lat-
tices; Sonia L’Innocente, Ultraproducts and Lie Algebras: Some Possible Interac-
tions; Ya-Qing Li, Defining New Generalized Functions by Nonstandard Discrete
Functions and Difference Quotients; Paolo Lipparini, Variations on Properties of
Ultrafilters Related to Compactness of Topological Spaces and to Model-Theoretical
Principles; Daniar Mushtari, Ultraproducts and Contiguity; Richard O’Donovan,
Analysis Using Relative Infinitesimals; Serpil Pehlivan, Weak Ideal Convergence
in Banach Spaces; Aleksandar Perovic, A Remark on Ultrapower Cardinality and
the Continuum Problem; Igor Protasov, Dynamical Equivalence on G∗; Yves Ray-
naud, Ultraproducts and Characterization of Certain Classes of Classical Banach
Lattices; Ireneusz Recl(cid:3)aw, Ideal Limits of Sequences of Continuous Functions and
a Game; David Ross, Automatic Continuity of Nonstandard Measures; Denis I.
Saveliev, On Hindman Sets; Christian Serp´e, Nonstandard Methods in Algebraic
Geometry; Jo˜ao Paulo Teixeira, Local-in-Time Existence and Regularity of the n-
Dimensional Navier-Stokes Equations via Discretizations; M. Akbari Tootkaboni,
Comfort Order on Locally Compact Topological Spaces; Boaz Tsaban, Super Ultra-
filters and Ramsey Theory; Hans Vernaeve, Algebras of Generalized Functions and
Nonstandard Analysis.