Terence Tao AMERICAN MATHEMATICAL SOCIETY Compactness and Contradiction

Terence Tao - American Mathematical Society · Compactness and Contradiction Terence Tao Compactness and Contradiction AMERICAN MATHEMATICAL SOCIETY

Embed Size (px)

Citation preview

Terence Tao


Compactness and Contradiction

Compactness and Contradiction

Terence Tao

Compactness and Contradiction


Compactness and Contradiction

Terence Tao



2010 Mathematics Subject Classification. Primary 00B15.

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-81

Library of Congress Cataloging-in-Publication Data

Library of Congress Cataloging-in-Publication Data has been applied for by the AMS.

See www.loc.gov/publish/cip/

ISBN: 978-0-8218-9492-7

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made bye-mail to [email protected].

c© 2013 Terence Tao. All rights reserved.Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13

To Garth Gaudry, who set me on the road;

To my family, for their constant support;

And to the readers of my blog, for their feedback and contributions.


Preface xi

A remark on notation xi

Acknowledgments xii

Chapter 1. Logic and foundations 1

§1.1. Material implication 1

§1.2. Errors in mathematical proofs 2

§1.3. Mathematical strength 4

§1.4. Stable implications 6

§1.5. Notational conventions 8

§1.6. Abstraction 9

§1.7. Circular arguments 11

§1.8. The classical number systems 12

§1.9. Round numbers 15

§1.10. The “no-self-defeating object” argument, revisited 16

§1.11. The “no-self-defeating object” argument, and the vaguenessparadox 28

§1.12. A computational perspective on set theory 35

Chapter 2. Group theory 51

§2.1. Torsors 51

§2.2. Active and passive transformations 54

§2.3. Cayley graphs and the geometry of groups 56

§2.4. Group extensions 62


viii Contents

§2.5. A proof of Gromov’s theorem 69

Chapter 3. Analysis 79

§3.1. Orders of magnitude, and tropical geometry 79

§3.2. Descriptive set theory vs. Lebesgue set theory 81

§3.3. Complex analysis vs. real analysis 82

§3.4. Sharp inequalities 85

§3.5. Implied constants and asymptotic notation 87

§3.6. Brownian snowflakes 88

§3.7. The Euler-Maclaurin formula, Bernoulli numbers, the zetafunction, and real-variable analytic continuation 88

§3.8. Finitary consequences of the invariant subspace problem 104

§3.9. The Guth-Katz result on the Erdos distance problem 110

§3.10. The Bourgain-Guth method for proving restriction theorems 123

Chapter 4. Non-Standard analysis 133

§4.1. Real numbers, non-standard real numbers, and finite precisionarithmetic 133

§4.2. Non-Standard analysis as algebraic analysis 136

§4.3. Compactness and contradiction: the correspondence principlein ergodic theory 137

§4.4. Non-Standard analysis as a completion of standard analysis 150

§4.5. Concentration compactness via non-standard analysis 168

Chapter 5. Partial differential equations 181

§5.1. Quasilinear well-posedness 181

§5.2. A type diagram for function spaces 189

§5.3. Amplitude-frequency dynamics for semilinear dispersiveequations 194

§5.4. The Euler-Arnold equation 203

Chapter 6. Miscellaneous 217

§6.1. Multiplicity of perspective 217

§6.2. Memorisation vs. derivation 220

§6.3. Coordinates 222

§6.4. Spatial scales 227

§6.5. Averaging 228

§6.6. What colour is the sun? 231

Contents ix

§6.7. Zeno’s paradoxes and induction 232

§6.8. Jevons’ paradox 233

§6.9. Bayesian probability 236

§6.10. Best, worst, and average-case analysis 242

§6.11. Duality 244

§6.12. Open and closed conditions 246

Bibliography 249

Index 255


In February of 2007, I converted my “What’s new” web page of researchupdates into a blog at terrytao.wordpress.com. This blog has since grownand evolved to cover a wide variety of mathematical topics, ranging from myown research updates, to lectures and guest posts by other mathematicians,to open problems, to class lecture notes, to expository articles at both basicand advanced levels. In 2010, I also started writing shorter mathematicalarticles on my Google Buzz feed at

profiles.google.com/114134834346472219368/buzz .

This book collects some selected articles from both my blog and my Buzzfeed from 2010, continuing a series of previous books [Ta2008], [Ta2009],[Ta2009b], [Ta2010], [Ta2010b], [Ta2011], [Ta2011b], [Ta2011c] basedon the blog.

The articles here are only loosely connected to each other, although manyof them share common themes (such as the titular use of compactness andcontradiction to connect finitary and infinitary mathematics to each other).I have grouped them loosely by the general area of mathematics they pertainto, although the dividing lines between these areas is somewhat blurry, andsome articles arguably span more than one category. Each chapter is roughlyorganised in increasing order of length and complexity (in particular, the firsthalf of each chapter is mostly devoted to the shorter articles from my Buzzfeed, with the second half comprising the longer articles from my blog).

A remark on notation

For reasons of space, we will not be able to define every single mathematicalterm that we use in this book. If a term is italicised for reasons other than


xii Preface

emphasis or for definition, then it denotes a standard mathematical object,result, or concept, which can be easily looked up in any number of references.(In the blog version of the book, many of these terms were linked to theirWikipedia pages, or other on-line reference pages.)

I will, however, mention a few notational conventions that I will usethroughout. The cardinality of a finite set E will be denoted |E|. We willuse the asymptotic notation X = O(Y ), X � Y , or Y � X to denote theestimate |X| ≤ CY for some absolute constant C > 0. In some cases we willneed this constant C to depend on a parameter (e.g., d), in which case weshall indicate this dependence by subscripts, e.g., X = Od(Y ) or X �d Y .We also sometimes use X ∼ Y as a synonym for X � Y � X.

In many situations there will be a large parameter n that goes off toinfinity. When that occurs, we also use the notation on→∞(X) or simplyo(X) to denote any quantity bounded in magnitude by c(n)X, where c(n)is a function depending only on n that goes to zero as n goes to infinity. Ifwe need c(n) to depend on another parameter, e.g., d, we indicate this byfurther subscripts, e.g., on→∞;d(X).

Asymptotic notation is discussed further in Section 3.5.

We will occasionally use the averaging notation

Ex∈Xf(x) :=1



to denote the average value of a function f : X → C on a non-empty finiteset X.

If E is a subset of a domain X, we use 1E : X → R to denote theindicator function of X, thus 1E(x) equals 1 when x ∈ E and 0 otherwise.


I am greatly indebted to many readers of my blog and buzz feed, includingRex Cheung, Dan Christensen, David Corfield, Quinn Culver, Tim Gow-ers, Greg Graviton, Zaher Hani, Bryan Jacobs, Bo Jacoby, Sune KristianJakobsen, Allen Knutson, Ulrich Kohlenbach, Diego Maldona, Mark Meckes,David Milovich, Timothy Nguyen, Michael Nielsen, Matthew Petersen, An-thony Quas, Pedro Lauridsen Ribeiro, Jason Rute, Americo Tavares, WillieWong, Qiaochu Yuan, Pavel Zorin, and several anonymous commenters, forcorrections and other comments, which can be viewed online at


The author is supported by a grant from the MacArthur Foundation, byNSF grant DMS-0649473, and by the NSF Waterman award.


[Ar1966] V. Arnold, Sur la geometrie differentielle des groupes de Lie de dimension in-finie et ses applications a l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier(Grenoble) 16 1966 fasc. 1, 319–361.

[ArKh1998] V. Arnold, B. Khesin, Topological methods in hydrodynamics. (English sum-mary) Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998.

[AuTa2010] T. Austin, T. Tao, On the testability and repair of hereditary hypergraphproperties, Random Structures and Algorithms 36 (2010), 373–463.

[AvGeTo2010] J. Avigad, P. Gerhardy, H. Towsner, Local stability of ergodic averages,Trans. Amer. Math. Soc. 362 (2010), no. 1, 261–288.

[BeCaTa2006] J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeyaconjectures, Acta Math. 196 (2006), no. 2, 261–302.

[BeHoMcPa2000] V. Bergelson, B. Host, R. McCutcheon, F. Parreau, Aspects of unifor-mity in recurrence. Colloq. Math. 84/85 (2000), part 2, 549–576.

[BeTaZi2010] V. Bergelson, T. Tao, T. Ziegler, An inverse theorem for the uniformityseminorms associated with the action of Fp, Geom. Funct. Anal. 19 (2010), no. 6,1539–1596.

[Bo1991] J. Bourgain, Besicovitch type maximal operators and applications to Fourieranalysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187.

[Bo1995] J. Bourgain, Estimates for cone multipliers, Geometric aspects of functionalanalysis (Israel, 1992–1994), 41–60, Oper. Theory Adv. Appl., 77, Birkhauser, Basel,1995.

[BoGu2010] J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on mul-tilinear estimates, Geom. Funct. Anal., 21 (2011), no. 6, 1239–1295.

[BoSm1975] J. Bona, R. Smith, The initial-value problem for the Korteweg-de Vries equa-tion, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601.

[CeMaPeRa2003] H. Cendra, J. Marsden, S. Pekarsky, T. Ratiu, Variational principlesfor Lie-Poisson and Hamilton-Poincare equations, Dedicated to Vladimir IgorevichArnold on the occasion of his 65th birthday. Mosc. Math. J. 3 (2003), no. 3, 833-867,1197–1198.


250 Bibliography

[ChSzTr1992] F. Chung, E. Szemeredi, W. T. Trotter, The number of different distancesdetermined by a set of points in the Euclidean plane, Discrete Comput. Geom. 7(1992), no. 1, 1–11.

[CoMi1997] T. Colding, W. Minicozzi, II. Harmonic functions on manifolds, Ann. Math.146 (1997), 725–747.

[Co1982] A. Cordoba, Geometric Fourier analysis, Ann. Inst. Fourier (Grenoble) 32(1982), no. 3, vii, 215–226.

[Co1985] A. Cordoba, Restriction lemmas, spherical summation, maximal functions,square functions and all that, Recent Progress in Fourier Analysis (El Escorial, 1983),57–64, North-Holland Math. Stud., 111, North-Holland, Amsterdam, 1985,

[De1912] M. Dehn, Transformation der Kurven auf zweiseitigen Flachen, Math. Ann. 72(1912), no. 3, 413–421.

[Dv2009] Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22(2009), no. 4, 1093–1097.

[EbMa1970] D. Ebin, J. Marsden, Groups of diffeomorphisms and the motion of an in-compressible fluid, Ann. of Math. 92 (1970) 102–163.

[ElSz2007] G. Elek, B. Szegedy, Limits of Hypergraphs, Removal and Regularity Lemmas.A Non-standard Approach, arXiv:0705.2179

[ElSh2010] G. Elekes, M. Sharir, Incidences in three dimensions and distinct distances inthe plane, Computational geometry (SCG’10), 413–422, ACM, New York, 2010.

[Er1946] P. Erdos, On sets of distances of n points, Amer. Math. Monthly 53 (1946),248–250.

[FrVi1990] S. Friedlander, M. Vishik, Lax pair formulation for the Euler equation, Phys.Lett. A 148 (1990), no. 6-7, 313–319.

[Ge1936] G. Gentzen, Die Widerspruchfreiheit der reinen Zahlentheorie, MathematischeAnnalen 112 (1936), 493–565.

[Go2008] W. T. Gowers, How can one equivalent statement be stronger than another?,gowers.wordpress.com/2008/12/28

[Go2010] W. T. Gowers, Decompositions, approximate structure, transference, and theHahn-Banach theorem, Bull. Lond. Math. Soc. 42 (2010), no. 4, 573–606.

[Gr2005] B. Green, Roth’s theorem in the primes, Ann. of Math. (2) 161 (2005), no. 3,1609–1636.

[GrTa2008] B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions,Annals of Math. 167 (2008), 481–547.

[Gr1981] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes

Etudes Sci. Publ. Math. No. 53 (1981), 53–73.

[Gu2010] L. Guth, The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya con-jecture, Acta Math. 205 (2010), no. 2, 263–286.

[GuKa2010] L. Guth, N. Katz, Algebraic methods in discrete analogs of the Kakeya prob-lem, Adv. Math. 225 (2010), no. 5, 2828–2839.

[GuKa2010b] L. Guth, N.H. Katz, On the Erdos distinct distance problem in the plane,arXiv:1011.4105v1 [math.CO]

[HeJeKoSt2009] S. Herrmann, A. Jensen, M. Joswig, B. Sturmfels, How to draw tropicalplanes, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of AndersBjorner, Research Paper 6, 26 pp.

Bibliography 251

[KaTa2004] N. Katz, G. Tardos, A new entropy inequality for the Erdos distance problem,Towards a theory of geometric graphs, 119–126, Contemp. Math., 342, Amer. Math.Soc., Providence, RI, 2004.

[KiVa2008] R. Killip, M. Visan, Nonlinear Schrodinger Equations at Critical Regularity,www.math.ucla.edu/∼visan/ClayLectureNotes.pdf

[Kl2010] B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth, J.Amer. Math. Soc. 23 (2010), no. 3, 815–829.

[KoSc1997] N. Korevaar, R. Schoen, Global existence theorems for harmonic maps to non-locally compact spaces, Comm. Anal. Geom. 5 (1997), no. 2, 333–387.

[LaPi2011] M. Larsen, R. Pink, Finite Subgroups of Algebraic Groups, J. Amer. Math.Soc. 24 (2011), no. 4, 1105–1158.

[Li2001] Y. Li, A Lax pair for the two dimensional Euler equation, J. Math. Phys. 42(2001), no. 8, 3552–3553.

[Lo1955] J. �Los, Quelques remarques, theoremes et problemes sur les classes definissablesd’algebres, in: Mathematical Interpretation of Formal Systems, North-Holland, Am-sterdam, 1955, 98–113.

[MaRa1999] J. Marsden, T. Ratiu, Introduction to mechanics and symmetry, A basicexposition of classical mechanical systems. Second edition. Texts in Applied Mathe-matics, 17. Springer-Verlag, New York, 1999.

[Mi1968] J. Milnor, Growth of finitely generated solvable groups, J. Diff. Geom. 2 (1968),447–449.

[Mo1995] N. Mok, Harmonic forms with values in locally compact Hilbert bundles, inProceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay 1993), SpecialIssue, 1995, pp. 433–454.

[Na1999] K. Nakanishi, Scattering theory for the nonlinear Klein-Gordon equation withSobolev critical power, Internat. Math. Res. Notices 1999, no. 1, 31–60.

[ReTrTuVa2008] O. Reingold, L. Trevisan, M. Tulsiani, S. Vadhan, New Proofs of theGreen-Tao-Ziegler Dense Model Theorem: An Exposition, preprint. arXiv:0806.0381

[Ro1953] K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 245-252.

[Ro2009] T. Roy, Global existence of smooth solutions of a 3D log-log energy-supercriticalwave equation, Anal. PDE 2 (2009), no. 3, 261–280.

[RuSz1978] I. Ruzsa, E. Szemeredi, Triple systems with no six points carrying three tri-angles, Colloq. Math. Soc. J. Bolyai 18 (1978), 939–945.

[Sa1915] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2,5th edition Hodges, Figgis and Co. Ltd. (1915).

[ScTi2002] I. Schindler, K. Tintarev, An abstract version of the concentration compactnessprinciple, Rev. Mat. Complut. 15 (2002), no. 2, 417–436.

[ShTa2010] Y. Shalom, T. Tao, A finitary version of Gromov’s polynomial growth theorem,Geom. Funct. Anal. 20 (2010), no. 6, 1502–1547.

[ShSt1998] J. Shatah, M. Struwe, Geometric wave equations. Courant Lecture Notes inMathematics, 2. New York University, Courant Institute of Mathematical Sciences,New York; American Mathematical Society, Providence, RI, 1998.

[Se1954] A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2),60 (1954), 365–374.

[Sh2011] H.-W. Shih, Some results on scattering for log-subcritical and log-supercriticalnonlinear wave equation, to appear, Analysis & PDE.

252 Bibliography

[St1979] E. M. Stein, Some problems in harmonic analysis. Harmonic analysis in Euclideanspaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part1, pp. 3–20, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence,R.I., 1979.

[Sz1975] E. Szemeredi, On sets of integers containing no k elements in arithmetic pro-gression, Acta Arith. 27 (1975), 299–345.

[Sz1978] E. Szemeredi, Regular partitions of graphs, in “Problemes Combinatoires etTheorie des Graphes, Proc. Colloque Inter. CNRS,” (Bermond, Fournier, Las Vergnas,Sotteau, eds.), CNRS Paris, 1978, 399–401.

[SzTr1873] E. Szemeredi, W. T. Trotter Jr., Extremal problems in discrete geometry, Com-binatorica 3 (1983), 381–392.

[Ta1951] A. Tarski, A decision method for elementary algebra and geometry, Universityof California Press, Berkeley and Los Angeles, Calif., 1951.

[Ta1999] T. Tao, The Bochner-Riesz conjecture implies the restriction conjecture, DukeMath. J. 96 (1999), no. 2, 363–375.

[Ta2003] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct.Anal. 13 (2003), no. 6, 1359–1384.

[Ta2003b] T. Tao, Recent progress on the Restriction conjecture, arXiv:math.CA/0311181

[Ta2004] T. Tao, A remark on Goldston-Yıldırım correlation estimates, unpublished.www.math.ucla.edu/∼tao/preprints/Expository/gy-corr.dvi

[Ta2006] T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in threespatial dimensions, Dyn. Partial Differ. Equ. 3 (2006), no. 2, 93–110.

[Ta2006b] T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS re-gional conference series in mathematics, 2006.

[Ta2006c] T. Tao, Analysis Vols. I, II, Hindustan Book Agency, 2006.

[Ta2007] T. Tao, Global regularity for a logarithmically supercritical defocusing nonlinearwave equation for spherically symmetric data, J. Hyperbolic Differ. Equ. 4 (2007), no.2, 259–265.

[Ta2007b] T. Tao, A correspondence principle between (hyper)graph theory and probabilitytheory, and the (hyper)graph removal lemma, J. d.Analyse Mathematique 103 (2007),1–45.

[Ta2008] T. Tao, Structure and Randomness: pages from year one of a mathematical blog,American Mathematical Society, Providence RI, 2008.

[Ta2008b] T. Tao, Norm convergence of multiple ergodic averages for commuting trans-formations, Ergodic Theory and Dynamical Systems 28 (2008), 657–688.

[Ta2009] T. Tao, Poincare’s Legacies: pages from year two of a mathematical blog, Vol.I, American Mathematical Society, Providence RI, 2009.

[Ta2009b] T. Tao, Poincare’s Legacies: pages from year two of a mathematical blog, Vol.II, American Mathematical Society, Providence RI, 2009.

[Ta2010] T. Tao, An epsilon of room, Vol. I, American Mathematical Society, ProvidenceRI, 2010.

[Ta2010b] T. Tao, An epsilon of room, Vol. II, American Mathematical Society, Provi-dence RI, 2010.

[Ta2011] T. Tao, An introduction to measure theory, American Mathematical Society,Providence RI, 2011.

[Ta2011b] T. Tao, Higher order Fourier analysis, American Mathematical Society, Provi-dence RI, 2011.

Bibliography 253

[Ta2011c] T. Tao, Topics in random matrix theory, American Mathematical Society, Prov-idence RI, 2011.

[TaVaVe1998] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction andKakeya conjectures, J. Amer. Math. Soc. 11 (1998), no. 4, 967–1000.

[TaVu2006] T. Tao, V. Vu, Additive combinatorics. Cambridge Studies in AdvancedMathematics, 105. Cambridge University Press, Cambridge, 2006.

[TaZi2010] T. Tao, T. Ziegler, The inverse conjecture for the Gowers norm over finitefields via the correspondence principle, Anal. PDE 3 (2010), no. 1, 1–20.

[TaZi2011] T. Tao, T. Ziegler, The inverse conjecture for the Gowers norm over finitefields in low characteristic, Ann. Comb. 16 (2012), no. 1, 121–188.

[Th1994] W. Thurston, On proof and progress in mathematics, Bull. Amer. Math. Soc.(N.S.) 30 (1994), no. 2, 161–177.

[Ti1972] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.

[To1975] P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math.Soc. 81 (1975), 477–478.

[Tz2006] N. Tzvetkov, Ill-posedness issues for nonlinear dispersive equations, Lectures onnonlinear dispersive equations, 63–103, GAKUTO Internat. Ser. Math. Sci. Appl., 27,Gakkotosho, Tokyo, 2006.

[Va1959] P. Varnavides, On certain sets of positive density, J. London Math. Soc. 39(1959), 358–360.

[Wo1968] J. Wolf, Growth of finitely generated solvable groups and curvature of Riemann-ian manifolds, J. Diff. Geom. 2 (1968), 421–446.

[Wo1995] T. Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat.Iberoamericana 11 (1995), no. 3, 651–674.

[Wo2001] T. Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153(2001), no. 3, 661–698.


a priori estimate, 185active transformation, 54Archimedean principle, 133Arzela-Ascoli diagonalisation trick, 139asymptotic notation, xii

Balog-Szemeredi-Gowers lemma, 230barrier, 107Bayes’ formula, 236Bayesian probability, 236Bernoulli numbers, 96Bolzano-Weierstrass theorem, 160Burgers’ equation, 185busy beaver function, 27

Cantor’s theorem, 21, 32Cartan-Killing form, 211Cayley graph, 57cell decomposition, 121characteristic subgroup, 67Christoffel symbols, 205cogeodesic flow, 205continuity method, 232coordinate system, 222

decomposition into varieties, 163difference equation, 186differentiating the equation, 185direct product, 66Duhamel’s formula, 182

elemengary convergence, 154energy, 197equipartition of energy, 203

Erdos distance problem, 110Euclid’s theorem, 19Euler equations of incompressible fluids,

213Euler-Arnold equation, 209Euler-Maclaurin formula, 98explicit formula, 102extension problem, 124

Faulhaber formula, 90finitely generated group, 57friendship paradox, 229Furstenberg correspondence principle,

137Furstenberg recurrence theorem, 143,


G-space, 51Godel incompleteness theorem, 25Godel sentence, 24Godel’s universe, 34Grandi’s series, 91Gromov’s theorem, 69, 140growth function, 105

harmonic function, 70Heine-Borel theorem, 161hereditary property, 68homogeneous space, 51

impredicativity of truth, 24indicator function, xiiinteresting number paradox, 31invariant subspace problem, 104


256 Index

Jordan’s theorem, 76

Klein geometry, 112Kleiner’s theorem, 70

lamplighter group, 52length contraction, 226Loeb measure, 165

mean ergodic theorem, 149metabelian group, 66metacyclic group, 66modus ponens, 239Morawetz inequality, 201

nilpotent group, 67non-standard universe, 158nonlinear wave equation, 194Notation, xinull hypothesis, 238

omnipotence paradox, 28oracle, 39overspill principle, 162

passive transformation, 54phase polynomial, 148Picard iteration, 181Poincare inequality, 75polycyclic group, 66polynomial ham sandwich theorem, 120problem of induction, 238product rule, 220profile decomposition, 174, 177

quasilinear equation, 183Quining trick, 24quotient rule, 220

regulus, 114restriction problem, 124

semi-direct product, 66semilinear equation, 182sequential Banach-Alaoglu theorem, 169Simpson’s paradox, 230smoothed sums, 91solvable group, 67sorites paradox, 30split exact sequence, 62standard part, 172stationary process, 142supersolvable group, 66Szemeredi regularity lemma, 166

Szemeredi’s theorem, 142, 164

Tarski’s undefinability theorem, 24torsor, 51tragedy of the commons, 235transfer principle, 159transport equation, 183trapezoidal rule, 94tropical algebra, 80Turing’s halting theorem, 26

ultrapower, 158underspill principle, 162uniquely transitive, 51universal set, 22

van der Waerden theorem, 163virtual properties, 68vorticity, 214vorticity equation, 214

wave equation, 194wave packet, 190word metric, 57

Zorn’s lemma, 34

��������������� ��������������� ��������� ���� ����� ���� �� � ���� ������� ���� ����� �������� �� � ���� ������������������� ��� ����������� ������ ����������� ��� ����� ��� ���������� �� ��� ���������� �� ������������ ��� ���������� �������� ��������� ����������� ��� ���� ��� ��� �� ���� ������� ����� ������������ ���� ������ � � ����� ����� �� �� ��� ������� ������ �������� ��� ����� ����� ��� ������� �� ���� ��������������������������������������������������������������������������

���� �� ������� ������� ��� � ��� �� ���� ����� ���� �������� ����� ��� �� ���!���� ���� ����� ����� �� "#$#�� % � �� ���� �� ������ ����� ��� ��� �� ���� ������&���� ����� ����� ���� ��� ������� ����� �������� � ��� ����� ����������� ���� �������������������������� �������������������� ����������� �����'�� ��������� ���������������� �������� ��������� ���� ���� ��������������������� ��� ���� ���������� ��������

(�� ����� �������� ����� ���!���������� �� ������������ ��������������������������������� ����� ���Graduate Studies in Mathematics��������


AMS on the Web www.ams.org

For additional informationand updates on this book, visit



d H