Advertisement

How I ‘met’ Dov Tamari

  • Jim Stasheff
Chapter
Part of the Progress in Mathematics book series (PM, volume 299)

Abstract

Although I never met Dov Tamari, neither in person nor electronically, our work had one important intersection – the associahedra. This Festschrift has given me the opportunity to set the record straight: the so-called Stasheff polytope was in fact constructed by Tamari in 1951, a full decade before my version. Here I will indulge in recollections of some of the history of the associahedra, its generalizations and applications. Others in this Festschrift will reveal still other aspects of Tamari’s vision, especially in more direct relation to the lattice/poset that bears his name.

Keywords

Open String Homotopy Type Loop Space Cluster Algebra Catalan Number 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.F. Adams, Infinite Loop Spaces, Princeton University Press, Princeton, 1978.Google Scholar
  2. 2.
    S. Axelrod and I.M. Singer, “Chern-Simons perturbation theory. II”, J. Differential Geom. 39 (1994) 173–213.MathSciNetzbMATHGoogle Scholar
  3. 3.
    L.C. Biedenharn, “An identity satisfied by the Racah coefficients”, J. Math. Physics 31 (1953) 287–293.MathSciNetzbMATHGoogle Scholar
  4. 4.
    R. Bott and C. Taubes, “On the self-linking of knots”, J. Math. Phys. 35 (1994) 5247–5287.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    V.M. Buchstaber, “Lectures on Toric Topology”, in Toric Topology Workshop, KAIST 2008, Trends inMathematics, Information Center of Mathematical Sciences, vol. 1, 2008, 1–55.Google Scholar
  6. 6.
    V.M. Buchstaber and V. Volodin, “Combinatorial 2-truncated cubes and applications”, in this volume. Google Scholar
  7. 7.
    J.W. Cannon and W.J. Floyd, “What is Thompson’s group”, Notices AMS (2011) 1112–1113.Google Scholar
  8. 8.
    J.S. Carter, D.E. Flath, and M. Saito, The classical and quantum 6j-symbols, Mathematical Notes, vol. 43, Princeton University Press, Princeton, NJ, 1995.Google Scholar
  9. 9.
    C. Ceballos, F. Santos, and G.M. Ziegler, “Many non-equivalent realizations of the associahedron”, arxiv.org/abs/1109.5544.
  10. 10.
    C. Ceballos and G.M. Ziegler, “Three non-equivalent realizations of the associahedron”, arxiv.org/abs/1006.3487.
  11. 11.
    S. Devadoss and S. Forcey, “Marked tubes and the graph multiplihedron”, Algebr. Geom. Topol. 8 (2008) 2081–2108.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    S. Devadoss, T. Heath, and C. Vipismakul, “Deformations of bordered surfaces and convex polytopes”, Notices of the American Mathematical Society 58 (2011) 530–541.MathSciNetzbMATHGoogle Scholar
  13. 13.
    V. Drinfel’d, “Quasi-Hopf algebras”, Leningrad Math J. 1 (1990) 1419–1457.MathSciNetzbMATHGoogle Scholar
  14. 14.
    F.J. Dyson, “Missed opportunities”, Bull. Amer. Math. Soc. 78 (1972) 635–652.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. Elliott, “Theoretical studies in nuclear spectroscopy V: The matrix elements of non-central forces with an application to the 2p-shell”, Proc. Roy. Soc. A 218 (1953) 345–370.CrossRefGoogle Scholar
  16. 16.
    Z. Fiedorowicz, S. Gubkin, and R. Vogt, “Associahedra and weak monoidal structures on categories”, arxiv.org/abs/1005.3979.
  17. 17.
    S. Forcey, “Convex Hull Realizations of the Multiplihedra”, Topology and its Applications 156 (2008) 326–347, arxiv.org/abs/0706.3226.
  18. 18.
    S. Forcey, “Quotients of the multiplihedron as categorified associahedra”, Homology, Homotopy Appl. 10 (2008) 227–256.MathSciNetzbMATHGoogle Scholar
  19. 19.
    K. Fukaya, “Floer homology, A∞-categories and topological field theory”, in Geometry and Physics, J. Andersen, J. Dupont, H. Pertersen, and A. Swan, eds., Lecture Notes in Pure and Applied Mathematics, vol. 184, Marcel-Dekker, 1995, Notes by P. Seidel, 9–32.Google Scholar
  20. 20.
    K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.Google Scholar
  21. 21.
    K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.Google Scholar
  22. 22.
    W. Fulton and R. MacPherson, “A compactification of configuration spaces”, Ann. Math. 139 (1994) 183–225.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    M. Haiman, “Constructing the associahedron”, available for download at http://math.berkeley.edu/~mhaiman/ftp/assoc/manuscript.pdf, 1984.
  24. 24.
    H. Hata, K. Itoh, T. Kugo, H. Kunitomo, and K. Ogawa, “Covariant string field theory”, Phys. Rev. D 34 (1986) 2360–2429.MathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Hatcher, Algebraic Topology, Cambridge Univeristy Press, 2002, available for download at http://www.math.cornell.edu/~hatcher/AT/ATpage.html..
  26. 26.
    C. Hohlweg and C.E.M.C. Lange, “Realizations of the associahedron and cyclohedron”, Discrete Comput. Geom. 37 (2007) 517–543.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    D. Huguet, “La structure du treillis des polyèdres de parenthésages”, Algebra Universalis 5 (1975) 82–87.MathSciNetCrossRefGoogle Scholar
  28. 28.
    D. Huguet and D. Tamari, “La structure polyédrale des complexes de parenthésages”, Journal of Combinatorics, Information & System Sciences 3 (1978) 69–81.MathSciNetzbMATHGoogle Scholar
  29. 29.
    N. Iwase and M. Mimura, “Higher homotopy associativity”, in Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, 193–220.Google Scholar
  30. 30.
    T. Kadeishvili, “On the homology theory of fibre spaces”, Russian Math. Surv. 35:3 (1980) 231–238, math.AT/0504 43 7.Google Scholar
  31. 31.
    M. Kaku and K. Kikkawa, “Field theorey of relativistic strings. I Trees”, Phys. Rev. D 10 (1974) 1110–1133.CrossRefGoogle Scholar
  32. 32.
    M.M. Kapranov, “The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation”, J. Pure and Appl. Alg. 85 (1993) 119–142.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    M. Kontsevich, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys. 66 (2003) 157–216.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    M.L. Laplaza, “Coherence for associativity not an isomorphism”, J. Pure Appl. Algebra 2 (1972) 107–120.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    R. Lashof, “Classification of fibre bundles by the loop space of the base”, Ann. of Math. (2) 64 (1956) 436–446.Google Scholar
  36. 36.
    C. Lee, “The associahedron and triangulations of the n-gon”, Europ. J. Combinatorics 10 (1989) 551–560.zbMATHGoogle Scholar
  37. 37.
    J.-L. Loday, “Realization of the Stasheff polytope”, Arch. Math. 83 (2004) 267–278.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    S. Mac Lane, “Natural associativity and commutativity”, Rice Univ. Studies 49 (1963) 28–46.MathSciNetGoogle Scholar
  39. 39.
    J.C. Moore, “Le théorème de Freudenthal, la suite exacte de James et l’invariant de Hopf généralisé”, in Séminaire Henri Cartan 7 (1954–1955) 22-01 – 22-15.Google Scholar
  40. 40.
    F. Muro and A. Tonks, “Unital associahedra”, arxiv.org/abs/1110.1959.
  41. 41.
    J. Morton, A. Shiu, L. Pachter, and B. Sturmfels, “The cyclohedron test for finding periodic genes in time course expression studies”, Stat. Appl. Genet. Mol. Biol. 6 (2007) Art. 21, 25 pp. (electronic).Google Scholar
  42. 42.
    U. Pachner, “P.L. homeomorphic manifolds are equivalent by elementary shellings”, European J. Combin. 12 (1991) 129–145.MathSciNetzbMATHGoogle Scholar
  43. 43.
    S. Saneblidze and R. Umble, “Matrads, Biassociahedra, and A-bialgebras”, Homology, Homotopy and Applications 13 (2011) 1–57.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    L. Shen, “Stasheff polytopes and the coordinate ring of the cluster algebra χ-variety A n”, arxiv.org/abs/1104.3528.
  45. 45.
    S. Shnider and J. Stasheff, “An operad-chik looks at configuration spaces, moduli spaces and mathematical physics, appendix B”, in Operads: Proceedings of Renaissance Conferences, J.-L. Loday, J. Stasheff, and A.A. Voronov, eds., Contemporary Mathematics, vol. 202, Amer. Math. Soc., 1996, 75–78.Google Scholar
  46. 46.
    D.D. Sleator, R.E. Tarjan, and W.P. Thurston, “Rotation distance, triangulations, and hyperbolic geometry”, J. Amer. Math. Soc. 1 (1988) 647–681.MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    J. Stasheff, “Homotopy associativity of H-spaces, I”, Trans. Amer. Math. Soc. 108 (1963) 293–312.MathSciNetGoogle Scholar
  48. 48.
    J.D. Stasheff, “The intrinsic bracket on the deformation complex of an associative algebra”, JPAA 89 (1993) 231–235, Festschrift in Honor of Alex Heller.Google Scholar
  49. 49.
    J. Stasheff, H-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin, 1970.Google Scholar
  50. 50.
    M. Sugawara, “On a condition that a space is group-like”, Math. J. Okayama Univ. 7 (1957) 123–149.MathSciNetzbMATHGoogle Scholar
  51. 51.
    M. Sugawara, “On the homotopy-commutativity of groups and loop spaces”, Mem. Coll. Sci. Univ. Kyoto, Ser. A Math. 33 (1960/61) 257–269.Google Scholar
  52. 52.
    D. Tamari, “Monoides préordonnés et chaînes de Malcev”, Doctorat ès-Sciences Mathématiques Thèse de Mathématique, Paris, 1951.Google Scholar
  53. 53.
    D. Tamari, “The algebra of bracketings and their enumeration”, Nieuw Archiefvoor Wiskunde 10 (1962) 131–146.Google Scholar
  54. 54.
    E.P. Wigner, “On the matrices which reduce the Kronecker products of representations of S. R. groups”, in Quantum Theory of Angular Momentum, Academic Press, New York, 1965, 87–133.Google Scholar
  55. 55.
    Wikipedia, “CW complex – Wikipedia, the free encyclopedia”, 2011, [Online; accessed 9- September-2011].Google Scholar
  56. 56.
    E. Witten, “Noncommutative geometry and string field theory”, Nuclear Phys. B 268 (1986) 253–294.MathSciNetCrossRefGoogle Scholar
  57. 57.
    G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995.Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.UNC-CHChapel HillUSA
  2. 2.U PennPhiladelphiaUSA

Personalised recommendations