Advertisement

Milnor’s Work in Algebra and Its Ramifications

  • Hyman BassEmail author
Chapter
Part of the The Abel Prize book series (AP)

Abstract

Highlights of Milnor’s extensive work in algebra are surveyed, with an emphasis on their ramifications and influence on further developments in the field. After brief discussion of his work on Hopf algebras, and on Growth of groups, more substantial treatment is given of his work on The Congruence Subgroup Problem, and on Algebraic K-theory and quadratic forms.

Keywords

Hopf Algebra Algebraic Group Finite Index Polynomial Growth Congruence Subgroup 
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.

Supplementary material

A lecture by Prof. Ghys in connection with the Abel Prize 2011 to John Milnor (MP4 399 MB)

A lecture by Prof. Hopkins in connection with the Abel Prize 2011 to John Milnor (MP4 306 MB)

A lecture by Prof. McMullen in connection with the Abel Prize 2011 to John Milnor (MP4 362 MB)

The Abel Lecture by John Milnor, the Abel Laureate 2011 (MP4 379 MB)

References

  1. 1.
    Arason, J., Pfister, A.: Beweis des Krullschen Durchschnittsatzes für den Wittring. Invent. Math. 12, 173–176 (1971) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bass, H., Lazard, M., Serre, J.-P.: Sous-groupes d’indices finis dans \(\operatorname{SL}(n,Z)\). Bull. Am. Math. Soc. 70, 385–392 (1964) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bass, H., Milnor, J., Serre, J.-P.: Solution of the congruence subgroup problem for \(\operatorname{SL}_{n}\) (n≥3) and \(\operatorname{Sp}_{2n}\) (n≥2). Publ. Math. IHES 33, 59–137 (1967) (Erratum: On a functorial property of power residue symbols. Publ. Math. IHES 44 (1974), 241–244) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bass, H.: Algebraic K-Theory. Benjamin, New York (1968) zbMATHGoogle Scholar
  5. 5.
    Bass, H.: On the degree of growth of a finitely generated nilpotent group. Proc. Lond. Math. Soc. 25, 603–614 (1972) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bass, H., Tate, J.: The Milnor ring of a global field. Springer LNM, vol. 342, pp. 349–446 (1973) zbMATHGoogle Scholar
  7. 7.
    Bass, H.: John Milnor the algebraist. In: Topological Methods in Modern Mathematics. A Symposium in Honor of John Milnor’s Sixtieth Birthday, pp. 45–83 (1993). Publish or Perish Google Scholar
  8. 8.
    Chevalley, C.: Deux théoèmes d’arithmétique. J. Math. Soc. Jpn. 3, 36–44 (1951) CrossRefGoogle Scholar
  9. 9.
    Grigorchuk, R.I.: On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271, 30–33 (1983) MathSciNetGoogle Scholar
  10. 10.
    Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. IHES 53, 53–73 (1981) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kato, K.: Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two. Invent. Math. 66(3), 493–510 (1982) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kneser, M.: Strong approximation, I, II. Algebraic groups and discontinuous subgroups. In: Proc. Symp. Pure Math. Amer Math. Soc., vol. IX, pp. 187–196 (1966) Google Scholar
  13. 13.
    Lubotzky, A.: Subgroup growth and congruence subgroups. Invent. Math. 119, 267–295 (1995) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Matsumoto, H.: Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. Éc. Norm. Super. 2, 1–62 (1969) CrossRefGoogle Scholar
  15. 15.
    Mennicke, J.: Finite factor groups of the unimodular group. Ann. Math. 81 (1965) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Merkurjev, A.S.: On the norm residue symbol of degree 2. Dokl. Akad. Nauk SSSR 261(3), 542–547 (1981) MathSciNetGoogle Scholar
  17. 17.
    Merkurjev, A.S.: Developments in algebraic K-theory and quadratic forms after the work of Milnor. In: Collected Papers of John Milnor, V. Algebra, pp. 399–417. Am. Math. Soc., Providence (2010) Google Scholar
  18. 18.
    Milnor, J.: The Steenrod algebra and its dual. Ann. of Math. (2) 67, 150–171 (1958) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Milnor, J., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81, 211–264 (1965) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Milnor, J.: Whitehead torsion. Bull. Amer. Math. Soc. (N. S.) 72, 358–426 (1966) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Milnor, J.: A note on curvature and the fundamental group. J. Differ. Geom. 2, 1–7 (1968) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geom. 2, 447–449 (1968) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Milnor, J.: Advanced problem 5603. MAA Monthly 75, 685–686 (1968) Google Scholar
  24. 24.
    Milnor, J.: Algebraic K-theory and quadratic forms. Invent. Math. 9, 318–344 (1969/1970) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Milnor, J.: Symmetric inner products in characteristic 2. Prospects in mathematics. In: Proc. Sympos., Princeton Univ., Princeton, NJ, 1970. Ann. of Math. Studies, vol. 70, pp. 59–75. Princeton Univ. Press, Princeton (1971) Google Scholar
  26. 26.
    Milnor, J.: Introduction to Algebraic K-Theory. Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971) zbMATHGoogle Scholar
  27. 27.
    Milnor, J.: Collected papers of John Milnor, v. In: Bass, H., Lam, T.-Y. (eds.) Algebra. Am. Math. Soc., Providence (2010) zbMATHGoogle Scholar
  28. 28.
    Moore, C.: Group extensions of p-adic and adèlic groups. Publ. Math. IHES 35, 5–70 (1968) CrossRefGoogle Scholar
  29. 29.
    Orlov, D., Vishik, A., Voevodsky, V.: An exact sequence for \(K_{\ast}^{M}/2\) with applications to quadratic forms. Ann. of Math. (2) 165(1), 1–13 (2007) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Platonov, V.P., Rapinchuk, A.S.: Abstract properties of S-arithmetic groups and the congruence subgroup problem. Russian Acad. Sci. Izv. Math. 40, 455–476 (1993) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Prasad, G., Rapinchuk, A.S.: Developments of the congruence subgroup problem after the work of Bass, Milnor, and Serre. In: Bass, H., Lam, T.-Y. (eds.) Collected Papers of John Milnor, V. Algebra. Am. Math. Soc., Providence (2010) Google Scholar
  32. 32.
    Quillen, D.: In: Higher Algebraic K-Theory I. Lecture Notes in Math., vol. 341, pp. 85–147. Springer, Berlin (1973) CrossRefGoogle Scholar
  33. 33.
    Raghunathan, M.S.: On the congruence subgroup problem. Publ. Math. IHES 46, 107–161 (1976) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Raghunathan, M.S.: On the congruence subgroup problem II. Invent. Math. 85, 73–117 (1986) MathSciNetCrossRefGoogle Scholar
  35. 35.
    Rost, M.: On Hilbert Satz 90 for K 3 for degree-two extensions, available as http://www.mathematik.uni-bielefeld.de/~rost/K3-86.html (1986)
  36. 36.
    Serre, J.-P.: Sur les groupes de congruence des variétés abéliennes. Izv. Akad. Nauk SSSR, Ser. Mat. 28, 3–18 (1964) MathSciNetzbMATHGoogle Scholar
  37. 37.
    Serre, J.-P.: Le probème des groupes de congruence pour \(\operatorname{SL}_{2}\). Ann. Math. 92, 489–527 (1970) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Serre, J.-P.: Sur les groupes de congruence des variétés abéliennes II. Izv. Akad. Nauk SSSR, Ser. Mat. 35, 731–735 (1971) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Steinberg, R.: Générateurs, relations et revêtements de groupes algébriques. In: Colloq. Théorie des Groupes Algébriques, Bruxelles, Librairie Universitaire, Louvain, 1962, pp. 113–127. Gauthier-Villars, Paris (1962) Google Scholar
  40. 40.
    Suslin, A., Voevodsky, V.: Bloch–Kato conjecture and motivic cohomology with finite coefficients. In: The Arithmetic and Geometry of Algebraic Cycles, Banff, AB, 1998. NATO Sci. Ser. C Math. Phys. Sci., vol. 548, pp. 117–189. Kluwer Academic, Dordrecht (2000) CrossRefGoogle Scholar
  41. 41.
    Tits, J.: Free subgroups of linear groups. J. Algebra 20, 250–270 (1972) MathSciNetCrossRefGoogle Scholar
  42. 42.
    Voevodsky, V.: Triangulated categories of motives over a field. In: Cycles, Transfers, and Motivic Homology Theories. Ann. of Math. Stud., vol. 143, pp. 188–238. Princeton Univ. Press, Princeton (2000) zbMATHGoogle Scholar
  43. 43.
    Voevodsky, V.: Motivic cohomology with Z/2-coefficients. IHES Math. Publ. 98, 59–104 (2003) MathSciNetCrossRefGoogle Scholar
  44. 44.
    Voevodsky, V.: Reduced power operations in motivic cohomology. Publ. Math. IHES 98, 1–57 (2003) MathSciNetCrossRefGoogle Scholar
  45. 45.
    Wolf, J.A.: Growth of finitely generated solvable groups, and curvature of Riemannian manifolds. J. Differ. Geom. 2, 421–446 (1968) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Michigan Ann ArborAnn ArborUSA

Personalised recommendations