A Beautiful Mind
Page 59
4. Harold Kuhn, interview, 11.30.97.
5. See, for example, June Barrow-Green, Poincaré and the Three-Body Problem (Providence, R.I.: American Mathematical Society, 1977); also Kuhn, interview.
6. George Hinman, interview, 10.30.97.
7. John F. Nash, Jr., Les Prix Nobel 1994, op. cit.
8. See, for example, E. T. Bell, Men of Mathematics, op. cit., and Norman Levinson, “Wiener’s Life,” in “Norbert Wiener 1894–1964,” Bulletin of the American Mathematical Society; vol. 72, no. 1, part II, p. 8.
9. Martin Davis, interview, 2.6.96.
10. Norman Steenrod, letter of recommendation, 2.51, as quoted by Kuhn, introduction, “A Celebration of John F. Nash, Jr.,” op. cit.
11. John Nash, “Algebraic Approximations of Manifolds,” op. cit., p. 516.
12. Solomon Lefschetz, President’s Report, Princeton University Archives, 7.18.80.
13. Solomon Lefschetz, memorandum, 3.9.49, on Spencer’s appointment as visiting professor at Princeton in academic year 1948–49; Donald Spencer, interviews, 11.28.95 and 11.29.95.
14. Lefschetz, memorandum, 3.9.49.
15. Donald Clayton Spencer, Biography, 10.61, Princeton University Archives.
16. See, for example, “Analysis, Complex,” Encyclopaedia Britannica (1962).
17. Kodaira won the Fields in 1954; David C. Spencer, “Kunihiko Kodaira (1915–1997),” American Mathematical Monthly, 2.98.
18. Spencer won the Bôcher in 1947, Biography, op. cit.
19. Lefschetz, memorandum, 3.9.49.
20. Joseph Kohn, professor of mathematics, Princeton University, interview, 7.19.95.
21. Ibid. Also Phillip Griffiths, director, Institute for Advanced Study, interview, 5.26.95.
22. In his recommendation for Spencer’s appointment as visiting professor in 1949, Lefschetz remarks on his “warm and sympathetic personality.” Spencer had an unusual willingness to reach out to colleagues in trouble. He became deeply involved in helping Max Shiffman, a bright young mathematician at Stanford who was diagnosed with schizophrenia; John Moore, a mathematician who suffered a severe depression; and John Nash after Nash returned to Princeton in the early 1960s. See Spencer, op. cit.
23. Spencer, op. cit.
24. As slightly restated by Milnor, “A Nobel Prize for John Nash,” op. cit., p. 14.
25. Intersectional Nomination: Class Five; 1996 Election, John F. Nash, Jr.
26. Michael Artin, professor of mathematics, MIT, interview, 12.2.97.
27. See, for example, Michael Artin and Barry Mazur, “On Periodic Points,” Annals of Mathematics, no. 81 (1965), pp. 82–99. Milnor calls this an “important” application.
28. Barry Mazur, professor of mathematics, Harvard University, interview, 12.3.97.
29. Nash cites, for example, H. Seifert, “Algebraische Approximation von Mannigfaltigkeiten,” Math. Zeit, vol. 41 (1936), pp. 1–17.
30. Ibid.
31. Steenrod, letter, 2.51, as quoted by Kuhn, introduction, “A Celebration of John F. Nash, Jr.,” op. cit.
32. Spencer, op. cit.
33. Nash, as told to Harold Kuhn, private communication, 12.2.97. The subsequent Nash-Moser theorem has even more profound implications for celestial mechanics. See Chapter 30.
34. Albert W. Tucker, interview, 11.94. Nash still dabbled in game theory, perhaps partly to maintain his RAND connection. For example, he wrote “N-Person Games: An Example and a Proof,” RAND Memorandum, RM-615, June 4, 1951, as well as, with graduate students Martin Shubik and John Mayberry, “A Comparison of Treatments of a Duopoly Situation,” RAND Memorandum P-222, July 10, 1951.
35. Kuhn, interview.
36. Letter from Albert W. Tucker to Hassler Whitney, 4.5.55.
37. Artin supervised the honors calculus program, which, according to John Tate (interview, 6.29.97), he took very seriously. Later documents refer to Nash’s having been a poor teacher; the comments undoubtedly stem from his experiences in 1950–51.
38. “There is no doubt that the department should look towards keeping Milnor permanently as a member of our faculty,” Solomon Lefschetz, President’s Report, Princeton University Archives, 9.51.
39. Letter from A. W. Tucker to H. Whitney, op. cit.
40. William Ted Martin, professor of mathematics, MIT, interview, 9.7.95.
41. Letter from Albert W. Tucker to Marshall Stone, 2.26.51.
42. Nash told Kuhn that his desire to live in Boston played a role in his accepting the MIT position, Kuhn, personal communication, 7.97.
16: MIT
1. Lindsay Russell, interview, 1.14.96.
2. Patrick Corcoran, retired captain, Cambridge City Police, interview, 8.12.97.
3. Felix Browder, interview, 11.14.95.
4. Gian-Carlo Rota, professor of mathematics, MIT, interview, 10.29.94.
5. Paul A. Samuelson, professor of economics, MIT, interview, 11.94.
6. Harvey Burstein, former FBI agent who set up the campus police at MIT, interview, 7.3.97.
7. Samuelson, interview.
8. William Ted Martin, professor of mathematics, MIT, interview, 9.7.95.
9. Samuelson, interview.
10. Department of Physics, MIT, communication, 1.98.
11. Course catalog, MIT, various years.
12. Samuelson, interview.
13. Ibid.
14. Arthur Mattuck, professor of mathematics, MIT, e-mail, 6.23.97.
15. Joseph Kohn, professor of mathematics, Princeton University, interview, 7.25.95.
16. Samuelson, interview. See also Report to the President, MIT, various years.
17. Jerome Lettvin, professor of electrical engineering and bioengineering, MIT, interview, 7.25.97; Emma Duchane, interview, 6.26.97.
18. Samuelson, interview.
19. Gian-Carlo Rota, interview.
20. Hearing before Committee on Un-American Activities (HUAC), House of Representatives, Eighty-third Congress, First Session, Washington, D.C., April 22 and 23, 1953.
21. Samuelson, interview.
22. Martin, interview.
23. Ibid.
24. See, for example, Wiener’s obituary, New York Times, 3.19.64; Paul Samuelson, “Some Memories of Norbert Wiener,” 1964, Xerox provided by Samuelson; and Norbert Wiener, Ex-Prodigy (New York: Simon & Schuster, 1953) and I Am a Mathematician (New York: Simon & Schuster, 1956).
25. Samuelson, “Some Memories of Norbert Wiener,” op. cit.
26. Ibid.
27. Zipporah Levinson, interview, 9.11.95.
28. Samuelson, “Some Memories of Norbert Weiner,” op. cit.
29. Z. Levinson, interview.
30. Ibid.
31. Ibid.
32. Ibid.
33. Note from John Nash to N. Wiener, 11.17.52.
34. Letter from John Nash to Albert W. Tucker, 10.58.
35. Jerome Neuwirth, professor of mathematics, University of Connecticut at Storrs, interview, 5.21.97.
36. The sketch of Levinson is based on recollections of his widow, Zipporah Levinson; Arthur Mattuck; F. Browder, 11.2.95; Gian-Carlo Rota, 11.94; and many others. Also Kenneth Hoffman, Memorandum to President J. B. Wiesner, 3.14.74; William Ted Martin et al., obituary of Norman Levinson, 12.17.75.
37. HUAC, op. cit. See also Chapter 19.
38. Arthur Mattuck, “Norman Levinson and the Distribution of Primes,” address to MIT shareholders, 10.6.78.
17: Bad Boys
1. Donald J. Newman, professor of mathematics, Temple University, interview, 12.28.95; Leopold Flatto, Bell Laboratories, interview, 4.25.96.
2. Sigurdur Helgason, professor of mathematics, MIT, interview, 2.13.96.
3. Course catalog, MIT, various years.
4. Arthur Mattuck, interview, 11.7.95.
5. Robert Aumann, professor of mathematics, Hebrew University, interview, 6.25.95.
6. Joseph Kohn, interview, 7.19.95.
7. Ibid.
8. Aumann, interview.
9. Sevmour Haber, professor
of mathematics, Temple University, interviews, 3.14.95 and 3.19.95.
10. George Whitehead, professor of mathematics, MIT, interview, 12.12.95.
11. Eva Browder, interview, 9.6.97.
12. Barry Mazur, interview, 12.3.97.
13. Harold Kuhn quotes Nash taking credit for introducing the tea hour at MIT in his introduction to the special volume in honor of Nash, “A Celebration of John F. Nash, Jr.,” op. cit.
14. Isadore M. Singer, professor of mathematics, MIT, interview, 12.13.95.
15. Kohn, interview.
16. Singer, interview.
17. Jerome Neuwirth, interview, 5.21.97.
18. Mattuck, interview, 2.13.96.
19. Descriptions of this legendary crowd are based on interviews with Kohn; Felix Browder, 11.2.95, 11.10.95, 9.6.97; Aumann; Neuwirth; Newman; H. F. Mattson, 10.29.97 and 11.18.97; Larry Wallen, 5.16.97 and 5.20.97; Mattuck; Paul Cohen, 1.5.96; Jacob Bricker, 5.22.97; and others.
20. F. Browder, interview, 9.6.97.
21. Haber, interview.
22. Ibid.
23. Martha Nash Legg, interview, 3.29.96.
24. Neuwirth, interview.
25. Ibid.
26. Mattuck, interview, 2.13.96.
27. Interviews with Neuwirth and F. Browder, 11.2.95.
28. Jürgen Moser, professor of mathematics, Eidgenossische Techische Hochschule, Zurich, interview, 3.23.96.
29. Marvin Minsky, professor of science, MIT, interview, 2.13.96.
30. Herta Newman, interview, 3.2.96.
31. Andrew Browder, professor of mathematics, Brown University, interview, 6.18.97.
32. Haber, interview.
33. Flatto, interview.
34. D. Newman, interview, 2.4.96.
35. Zipporah Levinson, interview, 9.11.95.
36. Neuwirth, interview.
37. D. Newman, interview.
38. Ibid.
39. Lawrence Wallen, professor of mathematics, University of Hawaii, interviews, 5.20.97 and 6.4.97.
40. Kohn, interview.
41. H. F. Mattson, professor of computer science, Syracuse University, interview, 5.16.97; also Wallen, interview.
42. J. C. Lagarias, “The Leo Collection: Anecdote and Stories,” AT&T Bell Laboratories, 4.29.95 (Xerox).
43. Mattuck, interview, 5.21.95, and Neuwirth, interview.
44. Neuwirth, interview.
45. The sketch of Donald J. Newman is based on an interview with him and on interviews with Flatto, Kohn, Mattuck, Singer, and Harold S. Shapiro, professor of mathematics, Royal Institute of Technology, Stockholm, Sweden, e-mail, 5.21.97.
46. Singer, interview, 12.13.95.
47. Mattuck, interview, 11.7.95.
48. D. Newman, interview, 3.2.96.
49. Helgason, interview, 12.3.94; also interviews with Mattuck and Singer.
50. Flatto, interview.
51. Ibid.
52. Ibid.
53. Singer, interview.
54. Haber, interview.
55. Ibid.
56. Flatto, interview.
57. Ibid.
58. Ibid.
59. Neuwirth, interview.
60. Ibid.
61. D. Newman, interview, 3.2.96.
62. Ibid.
63. H. Newman, interview.
64. Fred Brauer, professor of mathematics, University of Wisconsin, interview, 5.22.97.
18: Experiments
1. Harold N. Shapiro, professor of mathematics, Courant Institute, interview, 2.20.96.
2. John Milnor, interview, 9.26.95.
3. The account of the cross-country trip is based largely on recollections of Martha Nash Legg, interviews, 8.29.95 and 3.29.96, and Ruth Hincks Morgenson, interview, 6.22.97.
4. John Nash to Harold Kuhn, personal communication, 6.24.97; also Morgenson, interview.
5. M. Legg, interview.
6. Ibid.
7. Ibid.
8. Ibid.; Milnor, interview.
9. John M. Danskin, interview, 10.29.95.
10. M. Legg, interview.
11. Ibid.
12. John Milnor, “Games Against Nature,” in Decision Processes, edited by R. M. Thrall, C. H. Coombs, and R. L. Davis (New York: John Wiley & Sons, 1954).
13. “Some Games and Machines for Playing Them,” RAND Memorandum, D-l 164, 2.2.52.
14. John Nash and R. M. Thrall, “Some War Games,” RAND Memorandum, D-l379, 9.10.52.
15. G. Kalisch, J. Milnor, J. Nash, and E. Nering, “Some Experimental N-Person Games,” RAND Memorandum, RM-948, 8.25.52.
16. M. Legg, interview.
17. The description of the experiment is based on, apart from the original paper, Evar Nering, professor of mathematics, University of Minnesota, interview, 6.18.96; R. Duncan Luce and Howard Raiffa, Games and Decisions (New York: John Wiley & Sons, 1957), pp. 259–69; John H. Kagel and Alvin E. Roth, The Handbook of Experimental Economics, op. cit., pp. 10–11.
18. Kagel and Roth, op. cit.
19. Milnor, interview, 10.28.94.
20. John Milnor, “A Nobel Prize for John Nash,” op. cit.
21. See, for example, Kagel and Roth, op. cit.
22. Milnor, interview, 1.27.98.
23. Letter from John Nash to John Milnor, 12.27.64.
19: Reds
1. Zipporah Levinson, interview, 9.11.95.
2. Hearing before Committee on Un-American Activities, House of Representatives, Washington, D.C., 4.22.53 and 4.23.53. Unless otherwise noted, all references to the hearing are based on this transcript.
3. David Halberstam, The Fifties, op. cit.
4. Letter from Harold W. Dodds, president, Princeton University, to Colonel S. R. Gerard, Screening Division, Western Industrial Personnel Security Board, 10.14.54, Princeton University Archives.
5. See, for example, F. David Peat, Infinite Potential: The Life and Times of David Bohm (Reading, Mass.: Addison Wesley, 1997).
6. Z. Levinson, interview.
7. Ibid. See also Felix Browder, interview, 11.10.95.
8. Z. Levinson, interview.
9. Ibid.
10. The Tech, spring 1953, various issues.
11. Z. Levinson, interview.
12. Ibid.
13. William Ted Martin, interview.
14. Z. Levinson, interview.
15. Fred Brauer, e-mail, 6.23.97; Arthur H. Copeland, professor of mathematics, University of New Hampshire, e-mail, 6.24.97; Arthur Mattuck, e-mail, 6.25.97.
16. John Nash, plenary lecture, World Congress of Psychiatry, Madrid, 8.26.96, op. cit.
20: Geometry
1. Letter from Warren Ambrose to Paul Halmos, undated (written spring 1953).
2. The portrait of Ambrose is based on the recollections of Isadore Singer, 2.13.95; Lawrence Wallen, 6.4.97; Felix Browder, 11.2.95; Zipporah Levinson, 9.11.95; William Ted Martin, 9.7.95; H. F. Mattson, 10.29.97, 11.18.97, 11.28.97; Gian-Carlo Rota, 10.94; George Mackey, 12.14.95.
3. See, for example, I. M. Singer and H. Wu, “A Tribute to Warren Ambrose,” Notices of the AMS (April 1996).
4. Robert Aumann, interview, 6.28.95.
5. Gabriel Stolzenberg, professor of mathematics, Northeastern University, interview, 4.2.96.
6. Leopold Flatto, interview, 4.15.96. See also “The Leo Collection: Anecdotes and Stories,” AT&T Bell Laboratories, 4.29.94.
7. Ibid.
8. George Mackey, interview, 12.14.95.
9. Felix Browder, interview, 11.2.95.
10. Flatto, interview.
11. Despite its apocryphal ring, the story appears to be true and has been confirmed by Nash. Harold Kuhn, personal communication, 8.97.
12. Armand Borel, professor of mathematics, Institute for Advanced Study, interview, 3.1.96.
13. F. Browder, interview.
14. Ibid.
15. Joseph Kohn, interview, 7.19.95. Phrasing the question precisely, Ambrose would have used the adverb “isometrically” — meaning “to preserve distances” — after “embedding.”
16. Shlomo Sternberg, professor of mathematics, Harvard University, interview, 3.5.96.
17. Mikhail Gromov, interview. 12.16.97.
18. John Forbes Nash, Jr., Lcs Prix Nobel 1994, op. cit.
19. Gromov, interview.
20. John Conway, professor of mathematics, Princeton University, interview, 10.94.
21. Jürgen Moser, e-mail, 12.24.97.
22. Richard Palais, professor of mathematics, Brandeis University, interview, 11.6.95.
23. Moser, interview.
24. Donald J. Newman, interview, 3.2.96.
25. Jürgen Moser, “A Rapidly Convergent Iteration Method and Non-linear Partial Differential Equations, I, II,” Annali della Scuola Normale Superiore di Pisa, vol. 20 (1966), pp. 265–315, 499–535.
26. See, for example, Kyosi Ito, ed., Encyclopedic Dictionary of Mathematics (Mathematical Society of Japan; Cambridge: MIT Press, 1987), p. 1076; Lars Hörmander, “The Boundary Problems of Physical Geodesy,” Archive for Rational Mechanics and Analysis, vol. 62, no. 1 (1976), pp. 1–52; and S. Klainerman, Communications in Pure and Applied Mathematics, vol. 33 (1980), pp. 43–101.
27. John Nash, “C Isometric Imbeddings,” Annals of Mathematics, vol. 60, no. 3 (November 1954), pp. 383–96.
28. Kohn, interview.
29. John Forbes Nash, Jr., Les Prix Nobel 1994, op. cit.
30. Rota, interview, 11.14.95.
31. Flatto, interview.
32. Jacob Schwartz, professor of computer science, Courant Institute, interview, 1.29.96.
33. Isadore Singer, interview, 12.14.95.
34. Paul J. Cohen, professor of mathematics, Stanford University, interview, 1.6.96.
35. Moser, interview, 3.23.96.
36. The Nash-Federer correspondence wasn’t saved, and Federer declined to be interviewed (personal communication, 6.25.96). The account is based on the recollections of several individuals, including Wendell Fleming (interview, 6.97), a longtime collaborator and friend of Federer.
37. Fleming, interview.
38. John Nash, “The Imbedding Problem for Riemannian Manifolds,” Annuls of Mathematics, vol. 63, no. 1 (January 1956, received October 29, 1954, revised August 20, 1955).