Ultrafilters Across Mathematics: International Congress Ultramath 2008: Applications of Ultrafilters and Ultraproducts in Mathematics, June 1-7, 2008, Pisa, Italy PDF
Preview Ultrafilters Across Mathematics: International Congress Ultramath 2008: Applications of Ultrafilters and Ultraproducts in Mathematics, June 1-7, 2008, Pisa, Italy
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.
The list of books you might like
