Rough Structure and Classification

  • W. T. Gowers
Part of the Modern Birkhäuser Classics book series (MBC)


When I was first asked to speak at the “Visions in Mathematics” conference, I had what I believe was a typical reaction. I wanted to try to emulate Hilbert a century ago, but since I knew that I could not possibly match his breadth of vision, I was forced to make some sort of compromise. In this paper I shall discuss several open problems, not always in areas I know much about, but they are not intended as a list of the most important questions in mathematics, or even the most important questions in the areas of mathematics that I have worked in. Rather, they are a personal selection of problems that, for one reason or another, have captured my attention over the years.


Random Graph Travelling Salesman Problem Algebraic Number Arithmetic Progression Circuit Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AM]
    N. Alon, V.D. Milman, Eigenvalues, expanders and superconcentrators, Proc. 25th Annual FOCS, Singer Island, FL, IEEE, New York (1984), 320–322.Google Scholar
  2. [BW]
    R.P. Bambah, A.C. Woods, On a problem of Danzer, Pacific J. Math. 37 (1971), 295–301.zbMATHMathSciNetGoogle Scholar
  3. [B]
    F.A. Behrend, On sets of integers which contain no three in arithmetic progression, Proc. Nat. Acad. Sci. 23 (1946), 331–332.CrossRefMathSciNetGoogle Scholar
  4. [Bi]
    Y. Bilu, Structure of sets with small sumset, Astéerisque 258 (1999), 77–108.MathSciNetGoogle Scholar
  5. [Bol]
    J. Bourgain, Remarks on Montgomery’s conjectures on Dirichlet series, Geometric Aspects of Functional Analysis (1989–1990), Springer Lecture Notes in Mathematics 1469 (1991), 153–165.Google Scholar
  6. [Bo2]
    J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, GAFA 1 (1991), 147–187.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [Bo3]
    J. Bourgain, On triples in arithmetic progression, GAFA 9:5 (1999), 968–984.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [Bo4]
    J. Bourgain, On the dimension of Kakeya sets and related maximal in-equalities, GAFA 9:2 (1999), 256–282.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [Bu]
    George BUSH, Interview, Time, 26th Jan. 1987.Google Scholar
  10. [CG]
    F.R.K. Chung, R.L. GRAHAM, Quasi-random subsets of ℤn, J. Comb. Th. A 61 (1992), 64–86.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [CGW]
    F.R.K. Chung, R.L. Graham, R.M. Wilson, Quasi-random graphs, Combinatorica 9 (1989), 345–362.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [Co]
    S.A. Cook, The complexity of theorem proving procedures, Proc. 3rd Annual ACM Symposium on the Theory of Computing (1971), 151–158.Google Scholar
  13. [D]
    A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Symp. on Linear Spaces, Jerusalem (1961), 123–160.Google Scholar
  14. [E]
    P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294.CrossRefMathSciNetGoogle Scholar
  15. [ET]
    P. Erdős, P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.CrossRefGoogle Scholar
  16. [F]
    C. Fefferman, The multiplier problem for the ball, Annals of Math. 94 (1971), 330–336.CrossRefMathSciNetGoogle Scholar
  17. [FuK]
    H. Furstenberg, Y. Katznelson, A density version of the Hales-Jewett theorem, J. D’Analyse Math. 57 (1991), 64–119.zbMATHMathSciNetGoogle Scholar
  18. [Frl]
    G.A. Freiman, Foundations of a Structural Theory of Set Addition (in Russian), Kazan Gos. Ped. Inst., Kazan 1966.Google Scholar
  19. [Fr2]
    G.A. Freiman, Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs 37, Amer. Math. Soc, Providence, R.I.,USA, 1973.Google Scholar
  20. [Gl]
    W.T. Gowers, Lower bounds of tower type for Szemerédi’s uniformity lemma, GAFA 7 (1997), 322–337.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [G2]
    W.T. Gowers, A new proof of Szemerédi’s theorem for arithmetic progressions of length four, GAFA 8 (1998), 529–551.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [GrRR]
    R.L. Graham, V. Rödl, A. Rucinski, On graphs with linear Ramsey numbers, preprint.Google Scholar
  23. [H]
    W. Haken, Theorie der Normalflächen, ein Isotopiekriterium für den Kreisnoten, Acta Math. 105, 245–375.Google Scholar
  24. [HaW]
    J.M. Hammersley, D.J.A. Welsh, Further results on the rate of convergence to the connective constant of the hypercubical lattice, Quart. J. Math. Oxford Ser. (2) 13 (1962), 108–110.Google Scholar
  25. [HarSl]
    T. Hará, G. Slade, Self-avoiding walk in five or more dimensions, Comm. Math. Phys. 147 (1992), 101–136.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [HarS2]
    T. Hará, G. Slade, The lace expansion for self-avoiding walk in five or more dimensions, Rev. Math. Phys. 4 (1992), 235–327.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [He]
    D.R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. (2) 35 (1987), 385–394.Google Scholar
  28. [K]
    R.M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computations, Proc. Sympos., IBM Thomas J. Watson Res. Centr, Yorktown Heights, N.Y., 1972, (R.E. Miller, J.W. Thatcher, eds.) Plenum Press, New York 1972, 85–103.Google Scholar
  29. [KaLT]
    N.H. Katz, I. Laba, T. Tao, An improved bound on the Minkowski dimension of Besicovitch sets in ℝ3, Annals of Math., to appear.Google Scholar
  30. [KaT]
    N.H. Katz, T. Tao, A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture, submitted.Google Scholar
  31. [KoS]
    J. Komlós, M. Simonovits, Szemerédi’s Regularity Lemma and its applications in Graph Theory, in “Combinatorics, Paul Erdős is 80 (Vol 2)”, Bolyai Society Math. Studies 2, 295–352, Kesthely (Hungary) 1993, Budapest 1996.Google Scholar
  32. [LPS]
    A. Lubotzky, R. Phillips, P. Sarnak, Explicit expanders and the Ramanujan conjectures, Proceedings of the 18th ACM Symposium on the Theory of Computing 1986, 240–246; also Combinatorica 8 (1988), 261–277.Google Scholar
  33. [MS]
    N. Madras, G. Slade, The Self-Avoiding Random Walk, Birkhäuser, Boston, 1992.Google Scholar
  34. [Mi]
    V.D. Milman, A few observations on the connections between local theory and some other fields, in Geometric Aspects of Functional Analysis, Israel seminar (GAFA) 1986–1987 (J. Lindenstrauss, V.D. Milman, eds.), Springer LNM 1317, (1988), 283–289.Google Scholar
  35. [Mo]
    H.L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Math. 84, AMS 1994.Google Scholar
  36. [P]
    G. Polya, Mathematics and Plausible Reasoning, Vols. I and II, Princeton University Press, 1954.Google Scholar
  37. [RR]
    A.A. Razborov, S. Rudich, Natural proofs, in 26th Annual ACM Symposium on the Theory of Computing (STOC '94, Montreal, PQ, 1994); also J. Comput. System Sci. 55 (1997), 24–35.Google Scholar
  38. [Rol]
    K. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 245–252.CrossRefMathSciNetGoogle Scholar
  39. [Ro2]
    K. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20 (with corrigendum p. 168).CrossRefMathSciNetGoogle Scholar
  40. [Ru]
    I.Z. Ruzsa, Generalized arithmetic progressions and sumsets, Acta Math. Hungar. 65 (1995), 379–388.CrossRefMathSciNetGoogle Scholar
  41. [S1]
    E. Szemerédi, Regular partitions of graphs, Colloques Internationaux C.N.R.S. 260 — Problémes Combinatoires et Theorie des Graphes, Orsay 1976, 399–401.Google Scholar
  42. [S2]
    E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299–345.Google Scholar
  43. [S3]
    E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990), 155–158.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [TI]
    A.G. Thomason, Pseudo-random graphs, Proceedings of Random Graphs, Poznán 1985 (M. Karonski, ed.), Annals of Discrete Mathematics 33, 307–331.Google Scholar
  45. [T2]
    A.G. Thomason, A disproof of a conjecture of Erdős in Ramsey theory, J. London Math. Soc. 39 (1989), 246–255.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [T3]
    A.G. Thomason, Graph products and monochromatic multiplicities, Combinatorica 17 (1997), 125–134.CrossRefzbMATHMathSciNetGoogle Scholar
  47. [W]
    T.H. Wolff, An improved bound for Kakeya type maximal functions, Revista Mat. Iberoamericana 11 (1995), 651–674.zbMATHMathSciNetGoogle Scholar

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© Birkhäuser, Springer Basel AG 2010

Authors and Affiliations

  • W. T. Gowers
    • 1
  1. 1.Centre for Mathematical SciencesCambridgeUK

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