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Rough Structure and Classification

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Abstract

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.

Keywords

  • 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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References

  1. 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. R.P. Bambah, A.C. Woods, On a problem of Danzer, Pacific J. Math. 37 (1971), 295–301.

    MATH  MathSciNet  Google Scholar 

  3. F.A. Behrend, On sets of integers which contain no three in arithmetic progression, Proc. Nat. Acad. Sci. 23 (1946), 331–332.

    CrossRef  MathSciNet  Google Scholar 

  4. Y. Bilu, Structure of sets with small sumset, Astéerisque 258 (1999), 77–108.

    MathSciNet  Google Scholar 

  5. 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. J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, GAFA 1 (1991), 147–187.

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. J. Bourgain, On triples in arithmetic progression, GAFA 9:5 (1999), 968–984.

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. J. Bourgain, On the dimension of Kakeya sets and related maximal in-equalities, GAFA 9:2 (1999), 256–282.

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. George BUSH, Interview, Time, 26th Jan. 1987.

    Google Scholar 

  10. F.R.K. Chung, R.L. GRAHAM, Quasi-random subsets of ℤn, J. Comb. Th. A 61 (1992), 64–86.

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. F.R.K. Chung, R.L. Graham, R.M. Wilson, Quasi-random graphs, Combinatorica 9 (1989), 345–362.

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. 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. A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Symp. on Linear Spaces, Jerusalem (1961), 123–160.

    Google Scholar 

  14. P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294.

    CrossRef  MathSciNet  Google Scholar 

  15. P. Erdős, P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.

    CrossRef  Google Scholar 

  16. C. Fefferman, The multiplier problem for the ball, Annals of Math. 94 (1971), 330–336.

    CrossRef  MathSciNet  Google Scholar 

  17. H. Furstenberg, Y. Katznelson, A density version of the Hales-Jewett theorem, J. D’Analyse Math. 57 (1991), 64–119.

    MATH  MathSciNet  Google Scholar 

  18. G.A. Freiman, Foundations of a Structural Theory of Set Addition (in Russian), Kazan Gos. Ped. Inst., Kazan 1966.

    Google Scholar 

  19. 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. W.T. Gowers, Lower bounds of tower type for Szemerédi’s uniformity lemma, GAFA 7 (1997), 322–337.

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. W.T. Gowers, A new proof of Szemerédi’s theorem for arithmetic progressions of length four, GAFA 8 (1998), 529–551.

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. R.L. Graham, V. Rödl, A. Rucinski, On graphs with linear Ramsey numbers, preprint.

    Google Scholar 

  23. W. Haken, Theorie der Normalflächen, ein Isotopiekriterium für den Kreisnoten, Acta Math. 105, 245–375.

    Google Scholar 

  24. 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. T. Hará, G. Slade, Self-avoiding walk in five or more dimensions, Comm. Math. Phys. 147 (1992), 101–136.

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. T. Hará, G. Slade, The lace expansion for self-avoiding walk in five or more dimensions, Rev. Math. Phys. 4 (1992), 235–327.

    CrossRef  MATH  MathSciNet  Google Scholar 

  27. D.R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. (2) 35 (1987), 385–394.

    Google Scholar 

  28. 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. 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. 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. 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. 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. N. Madras, G. Slade, The Self-Avoiding Random Walk, Birkhäuser, Boston, 1992.

    Google Scholar 

  34. 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. 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. G. Polya, Mathematics and Plausible Reasoning, Vols. I and II, Princeton University Press, 1954.

    Google Scholar 

  37. 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. K. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 245–252.

    CrossRef  MathSciNet  Google Scholar 

  39. K. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20 (with corrigendum p. 168).

    CrossRef  MathSciNet  Google Scholar 

  40. I.Z. Ruzsa, Generalized arithmetic progressions and sumsets, Acta Math. Hungar. 65 (1995), 379–388.

    CrossRef  MathSciNet  Google Scholar 

  41. 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. E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299–345.

    Google Scholar 

  43. E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990), 155–158.

    CrossRef  MATH  MathSciNet  Google Scholar 

  44. 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. A.G. Thomason, A disproof of a conjecture of Erdős in Ramsey theory, J. London Math. Soc. 39 (1989), 246–255.

    CrossRef  MATH  MathSciNet  Google Scholar 

  46. A.G. Thomason, Graph products and monochromatic multiplicities, Combinatorica 17 (1997), 125–134.

    CrossRef  MATH  MathSciNet  Google Scholar 

  47. T.H. Wolff, An improved bound for Kakeya type maximal functions, Revista Mat. Iberoamericana 11 (1995), 651–674.

    MATH  MathSciNet  Google Scholar 

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Gowers, W.T. (2010). Rough Structure and Classification. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds) Visions in Mathematics. Modern Birkhäuser Classics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0422-2_4

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