The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)

NOTES

PRELUDE

1 Plato, Timaeus, trans. Donald J. Zeyl (Indianapolis: Hackett, 2000), p. 12.

2 Ibid., pp. 46-47.

3 Ibid., p. 44.

CHAPTER 1

1 Max Tegmark, interview with author, May 16, 2005. (Note: All interviews were conducted by Steve Nadis unless otherwise noted.)

2 Aristotle, On the Heavens, at Ancient Greek Online Library, http://greektexts.com/library/Aristotle/On_The_Heavens/eng/print/1043.html.

3 Michio Kaku, Hyperspace (New York: Anchor Books, 1995), p. 34.

4 H. G. Wells, The Time Machine (1898), available at http://www.bartleby.com/1000/1.html.

5 Abraham Pais, Subtle Is the Lord (New York: Oxford University Press, 1982), p. 152.

6 Oskar Klein, “From My Life of Physics,” in The Oskar Klein Memorial Lectures, ed. Gosta Ekspong (Singapore: World Scientific, 1991), p. 110.

7 Leonard Mlodinow, Euclid’s Window (New York: Simon & Schuster, 2002), p. 231.

8 Andrew Strominger, “Black Holes and the Fundamental Laws of Nature,” lecture, Harvard University, Cambridge, MA, April 4, 2007.

9 Ibid.

CHAPTER 2

1 Georg Friedrich Bernhard Riemann, “On the Hypotheses Which Lie at the Foundations of Geometry,” lecture, Göttingen Observatory, June 10, 1854.

2 E. T. Bell, Men of Mathematics (New York: Simon & Schuster, 1965), p. 21.

3 Leonard Mlodinow, Euclid’s Window (New York: Simon & Schuster, 2002), p. xi.

4 Edna St. Vincent Millay, “Euclid Alone Has Looked on Beauty Bare,” quoted in Robert Osserman, Poetry of the Universe (New York: Anchor Books, 1995), p. 6.

5 Andre Nikolaevich Kolmogorov, Mathematics of the 19th Century (Boston, Birkhauser, 1998).

6 Deane Yang (Polytechnic Institute of New York University), e-mail letter to author, April 20, 2009.

7 Mlodinow, Euclid’s Window, p. 205.

8 Brian Greene, The Elegant Universe (New York: Vintage Books, 2000), p. 231.

9 C. N. Yang, “Albert Einstein: Opportunity and Perception,” speech, 22nd International Conference on the History of Science, Beijing, China, 2005.

10 Chen Ning Yang, “Einstein’s Impact on Theoretical Physics in the 21st Century,” AAPPS Bulletin 15 (February 2005).

11 Greene, The Elegant Universe, p. 72.

CHAPTER 3

1 Robert Greene (UCLA), interview with author, March 13, 2008.

2 Lizhen Ji and Kefeng Liu, “Shing-Tung Yau: A Manifold Man of Mathematics,” Proceedings of Geometric Analysis: Present and Future Conference, Harvard University, August 27-September 1, 2008.

3 Leon Simon (Stanford University), interview with author, February 6, 2008.

4 Greene, interview with author, March 13, 2008.

5 Cameron Gordon (University of Texas), interview with author, March 14, 2008.

6 Robert Geroch (University of Chicago), interview with author, February 28, 2008.

7 Edward Witten (Institute for Advanced Study), interview with author, March 31, 2008.

8 Edward Witten, “A New Proof of the Positive Energy Theorem,” Communications in Mathematical Physics 80 (1981): 381-402.

9 Roger Penrose, “Gravitational Collapse: The Role of General Relativity,” 1969, reprinted in Mathematical Intelligencer 30 (2008): 27-36.

10 Richard Schoen (Stanford University), interview with author, January 31, 2008.

11 Demetrios Christodoulou, The Formation of Black Holes in General Relativity (Zurich: European Mathematical Society, 2009).

12 John D. S. Jones, “Mysteries of Four Dimensions,” Nature 332 (April 7, 1998): 488-489.

13 Simon Donaldson (Imperial College), interview with author, April 3, 2008.

14 Faye Flam, “Getting Comfortable in Four Dimensions,” Science 266 (December 9, 1994): 1640.

15 Ibid.

16 Mathematical Institute at the University of Oxford, “Chart the Realm of the 4th Dimension,” http://www2.maths.ox.ac.uk/~dusautoy/2soft/4D.htm.

17 Grisha Perelman, “The Entropy Formula for the Ricci Flow and Its Geometric Applications,” November 11, 2002, http://arxiv.org/abs/math/0211159v1.

CHAPTER 4

1 Eugenio Calabi (University of Pennsylvania), interview with author, October 18, 2007.

2 Robert Greene (UCLA), interview with author, October 18, 2008.

3 Calabi, interview with author, October 18, 2008.

4 Greene, interview with author, June 24, 2008.

5 Calabi, interview with author, October 18, 2007.

CHAPTER 5

1 Robert Greene (UCLA), interview with author, January 29, 2008.

2 Eugenio Calabi (University of Pennsylvania), interview with author, May 14, 2008.

3 Ibid.

4 Erwin Lutwak (Polytechnic Institute of NYU), interview with author, May 15, 2008.

5 Calabi, interview with author, May 14, 2008.

6 Calabi, interview with author, June 16, 2008.

7 Ibid.

8 Calabi, interview with author, October 18, 2007.

CHAPTER 6

1 Cumrun Vafa (Harvard University), interview with author, January 19, 2007.

2 John Schwarz (California Institute of Technology), interview with author, August 13, 2008.

3 Michael Green (University of Cambridge), e-mail letter to author, August 15, 2008.

4 Schwarz, interview with author, August 13, 2008.

5 Andrew Strominger (Harvard University), interview with author, February 7, 2007.

6 Strominger, interview with author, November 1, 2007.

7 Raman Sundrum (Johns Hopkins University), interview with author, January 25, 2007.

8 Strominger, interview with author, February 7, 2007.

9 Dennis Overbye, “One Cosmic Question, Too Many Answers,” New York Times, September 2, 2003.

10 Juan Maldacena (Princeton University), interview with author, September 9, 2007.

11 Dan Freed (University of Texas), interview with author, June 24, 2008.

12 Tristan Hubsch (Howard University), interview with author, August 30, 2008.

13 Gary Horowitz (University of California, Santa Barbara), interview with author, February 15, 2007.

14 Eugenio Calabi (University of Pennsylvania), interview with author, October 18, 2007.

15 Woody Allen, “Strung Out,” New Yorker, July 28, 2003.

16 Liam McAllister (Cornell University), e-mail letter to author, April 24, 2009.

17 Allan Adams (MIT), interview with author, August 10, 2007.

18 Joe Polchinski (University of California, Santa Barbara), interview with author, January 29, 2007.

19 Brian Greene, The Fabric of the Cosmos (New York: Alfred A. Knopf, 2004), p. 372.

20 P. Candelas, G. Horowitz, A. Strominger, and E. Witten, “Vacuum Configurations for Superstrings,” Nuclear Physics B 258 (1985): 46-74.

21 Edward Witten (IAS), e-mail letter to author, July 24, 2008.

22 Volker Braun, Philip Candelas, and Rhys Davies, “A Three-Generation Calabi-Yau Manifold with Small Hodge Numbers,” October 28, 2009, http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5464v1.pdf.

23 Dennis Overbye, “One Cosmic Question, Too Many Answers,” New York Times, September 2, 2003.

24 Dale Glabach and Juan Maldacena, “Who’s Counting?” Astronomy, (May 2006), 72.

25 Andrew Strominger, “String Theory, Black Holes, and the Fundamental Laws of Nature,” lecture, Harvard University, Cambridge, MA, April 4, 2007.

26 Witten, e-mail letter to author, July 21, 2008.

27 Petr Horava (University of California, Berkeley), interview with author, July 6, 2007.

28 Ibid.

29 Leonard Susskind (Stanford University), interview with author, May 25, 2007.

CHAPTER 7

1 Ronen Plesser (Duke University), interview with author, September 3, 2008.

2 Ibid.

3 Marcus Grisaru (McGill University), interview with author, August 18, 2008.

4 Plesser, interview with author, September 3, 2008.

5 Shamit Kachru (Stanford University), interview with author, August 19, 2008.

6 Ashoke Sen (Harish-Chandra Research Institute), interview with author, August 22, 2008.

7 Jacques Distler and Brian Greene, “Some Exact Results on the Superpotential from Calabi-Yau Compactifications,” Nuclear Physics B 309 (1988): 295-316.

8 Doron Gepner, “Yukawa Couplings for Calabi-Yau String Compactification,” Nuclear Physics B 311 (1988): 191-204.

9 Kachru, interview with author, August 19, 2008.

10 Paul Aspinwall (Duke University), interview with author, August 14, 2008.

11 Wolfgang Lerche, Cumrun Vafa, and Nicholas Warner, “Chiral Rings in N = 2 Superconformal Theories,” Nuclear Physics B 324 (1989): 427-474.

12 B. R. Greene, C. Vafa, and N. P. Warner, “Calabi-Yau Manifolds and Renormalization Group Flows,” Nuclear Physics B 324 (1989): 371-390.

13 Brian Greene (Columbia University), interview with author, March 11, 2010.

14 Ibid.

15 Doron Gepner, interview with author, August 19, 2008.

16 B. R. Greene and M. R. Plesser, “Duality in Calabi-Yau Moduli Space,” Nuclear Physics B 338 (1990): 15-37.

17 Brian Greene, The Elegant Universe (New York: Vintage Books, 2000), p. 258.

18 Plesser, interview with author, September 19, 2008.

19 Greene, interview with author, March 11, 2010.

20 Greene, The Elegant Universe, p. 259.

21 Cumrun Vafa (Harvard University), interview with author, September 19, 2008.

22 Greene, interview with author, March 13, 2010.

23 Mark Gross (UCSD), interview with author, October 31, 2008.

24 Andreas Gathmann (University of Kaiserslautern), interview with author, August 25, 2008.

25 David Hilbert, “Mathematical Problems,” lecture, International Congress of Mathematicians, Paris, 1900, http://aleph0.clarku.edu/~djoyce/hilbert/problems.html (html version prepared by David Joyce, Mathematics Department, Clark University, Worcester, MA).

26 Andreas Gathmann, “Mirror Principle I,” Mathematical Reviews, MR1621573, 1999.

27 David Cox (Amherst College), interview with author, June 13, 2008.

28 Andrew Strominger (Harvard University), interview with author, February 7, 2007.

29 Gross, interview with author, September 19, 2008.

30 Ibid.

31 Gross, interview with author, September 24, 2008.

32 Eric Zaslow (Northwestern University), interview with author, June 26, 2008.

33 A. Strominger, S. T. Yau, and E. Zaslow, “Mirror Symmetry Is T Duality,” Nuclear Physics 479 (1996): 243-259.

34 Gross, interview with author, September 24, 2008.

35 Mark Gross, e-mail letter to author, September 29, 2008.

36 Ibid.

37 Strominger, interview with author, August 1, 2007.

38 Zaslow, interview with author, June 26, 2008.

39 Gross, interview with author, September 19, 2008.

40 Yan Soibelman (Kansas State University), interview with author, September 26, 2008.

41 Aspinwall, interview with author, June 23, 2008.

42 Michael Douglas (Stony Brook University), interview with author, August 20, 2008.

43 Aspinwall, interview with author, June 23, 2008.

44 Gross, interview with author, September 24, 2008.

CHAPTER 8

1 Avi Loeb (Harvard University), interview with author, September 25, 2008.

2 American Mathematical Society, “Interview with Heisuke Hironaka,” Notices of the AMS 52, no. 9 (October 2005): 1015.

3 Steve Nadis, “Cosmic Inflation Comes of Age,” Astronomy (April 2002).

4 Andrew Strominger, “String Theory, Black Holes, and the Fundamental Laws of Nature,” lecture, Harvard University, Cambridge, MA, April 4, 2007.

5 Ibid.

6 Hirosi Ooguri (California Institute of Technology), interview with author, October 8, 2008.

7 Strominger, lecture.

8 Andrew Strominger and Cumrun Vafa, “Microscopic Origin of the Bekenstein-Hawking Entropy,” Physics Letters B 379 ( June 27, 1996): 99-104.

9 Andrew Strominger, quoted in Gary Taubes, “Black Holes and Beyond,” Science Watch, May/June 1999, http://archive.sciencewatch.com/may-june99/sw_may-june99_page3.htm.

10 Ooguri, interview with author, October 8, 2008.

11 Strominger, quoted in Taubes, “Black Holes and Beyond.”

12 Xi Yin (Harvard University), interview with author, October 14, 2008.

13 Ibid.

14 Yin, interview with author, October 22, 2008.

15 Frederik Denef (Harvard University), interview with author, August 26, 2008.

16 Yin, interview with author, October 14, 2008.

17 Aaron Simons, interview with author, February 9, 2007.

18 Ooguri, interview with author, October 8, 2008.

19 Simons, interview with author, February 9, 2007.

20 J. M. Maldacena, A. Strominger, and E. Witten, “Black Hole Entropy in M-Theory,” Journal of High Energy Physics 9712 (1997), http://arxiv.org/PS_cache/hep-th/pdf/9711/9711053v1.pdf.

21 Juan Maldacena (IAS), interview with author, September 4, 2008.

22 Hirosi Ooguri, Andrew Strominger, and Cumrun Vafa, “Black Hole Attractors and the Topological String,” Physical Review D 70 (2004).

23 Cumrun Vafa (Harvard University), interview with author, September 26, 2008.

24 James Sparks (Harvard University), interview with author, February 6, 2007.

25 Amanda Gefter, “The Elephant and the Event Horizon,” New Scientist (October 26, 2006): 36-39.

26 John Preskill, “On Hawking’s Concession,” July 24, 2004, http://www.theory.caltech.edu/~preskill/jp_24jul04.html.

27 Andrew Strominger (Harvard University), interview with author, February 7, 2007.

28 Juan Maldacena, “The Illusion of Gravity,” Scientific American, November 2005, pp. 57-58, 61.

29 Davide Castelvecchi, “Shadow World,” Science News 172 (November 17, 2007).

30 Taubes, “Black Holes and Beyond.”

CHAPTER 9

1 L. Frank Baum, The Wizard of Oz (Whitefish, MT: Kessinger, 2004), p. 111.

2 Volker Braun (Dublin Institute for Advanced Studies), interview with author, November 4, 2008.

3 Philip Candelas (Oxford University), interview with author, December 1, 2008.

4 Ibid.

5 Andrew Strominger (Harvard University), interview with author, February 7, 2007.

6 Cumrun Vafa, “The Geometry of Grand Unified Theories,” lecture, Harvard University, Cambridge, MA, August 29, 2008.

7 Chris Beasley (Stony Brook University), interview with author, November 13, 2008.

8 Burt Ovrut (University of Pennsylvania), interview with author, July 20, 2008.

9 Ovrut, interview with author, February 2, 2007.

10 Ron Donagi (University of Pennsylvania), interview with author, November 14, 2008.

11 Donagi, interview with author, November 19, 2008.

12 Candelas, interview with author, December 1, 2008.

13 Donagi, interview with author, May 3, 2008

14 Ovrut, interview with author, November 20, 2008.

15 Donagi, interview with author, November 20, 2008.

16 Ovrut, interview with author, November 20, 2008.

17 Shamit Kachru (Stanford University), interview with author, November 4, 2008.

18 Michael Douglas (Stony Brook University), interview with author, August 20, 2008.

19 Candelas, interview with author, December 1, 2008.

20 Simon Donaldson (Imperial College), interview with author, November 29, 2008.

21 Ovrut, interview with author, November 19, 2008.

22 Candelas, interview with author, December 1, 2008.

23 Strominger, interview with author, February 7, 2007.

24 Adrian Cho, “String Theory Gets Real—Sort Of,” Science 306 (November 26, 2004): 1461.

25 Candelas, interview with author, December 1, 2008.

26 Allan Adams (MIT), interview with author, November 15, 2008.

CHAPTER 10

1 Gary Shiu, quoted in Adrian Cho, “String Theory Gets Real—Sort Of,” Science 306 (November 26, 2004): 1461.

2 Shamit Kachru (Stanford University), e-mail letter to author, December 6, 2008.

3 Shamit Kachru, Renata Kallosh, Andrei Linde, and Sandip Trivedi, “De Sitter Vacua in String Theory,” Physical Review D 68 (2003).

4 Raman Sundrum (Johns Hopkins University), interview with author, February 22, 2007.

5 Liam McAllister (Cornell University), interview with author, November 12, 2008.

6 Kachru, interview with author, September 8, 2007.

7 McAllister (Princeton University), interview with author, February 20, 2007.

8 Joe Polchinski (University of California, Santa Barbara), interview with author, February 6, 2006.

9 David Gross, quoted in Dennis Overbye, “Zillions of Universes? Or Did Ours Get Lucky?” New York Times, October 28, 2003.

10 Burton Richter, “Randall and Susskind,” letter to editor, New York Times, January 29, 2006.

11 Leonard Susskind, The Cosmic Landscape (New York: Little, Brown, 2006), pp. 354-355.

12 Tristan Hubsch (Howard University), interview with author, November 7, 2008.

13 Mark Gross (UCSD), interview with author, October 31, 2008.

14 Gross, interview with author, September 19, 2008.

15 Miles Reid (University of Warwick), interview with author, August 12, 2007.

16 Allan Adams (MIT), interview with author, October 31, 2008.

17 Gross, interview with author, October 31, 2008.

18 Adams, interview with author, October 31, 2008.

19 Tristan Hubsch, e-mail letter to author, December 15, 2008.

20 Melanie Becker (Texas A&M University), interview with author, February 1, 2007.

21 Andrew Strominger (Harvard University), interview with author, February 7, 2007.

22 Li-Sheng Tseng (Harvard University), interview with author, December 17, 2008.

23 Becker, interview with author, February 1, 2007.

24 Polchinski, interview with author, January 29, 2007.

25 Strominger, interview with author, August 1, 2007.

26 Burt Ovrut (University of Pennsylvania), interview with author, February 2, 2007.

CHAPTER 11

1 Geoffrey Landis, “Vacuum States,” Asimov’s Science Fiction 12 ( July 1988): 73-79.

2 Andrew R. Frey, Matthew Lippert, and Brook Williams, “The Fall of Stringy de Sitter,” Physical Review D 68 (2003).

3 Sidney Coleman, “Fate of the False Vacuum: Semiclassical Theory,” Physical Review D 15 (May 15, 1977): 2929-2936.

4 Steve Giddings (University of California, Santa Barbara), interview with author, September 24, 2007.

5 Matthew Kleban (New York University), interview with author, January 17, 2008.

6 Dennis Overbye, “One Cosmic Question, Too Many Answers,” New York Times, September 2, 2003.

7 Andrei Linde (Stanford University), interview with author, December 27, 2007.

8 Giddings, interview with author, October 17, 2007.

9 Shamit Kachru (Stanford University), interview with author, September 18, 2007.

10 Linde, interview with author, December 27, 2007.

11 Henry Tye (Cornell University), interview with author, September 12, 2007.

12 Linde, interview with author, January 10, 2008.

13 S. W. Hawking, “The Cosmological Constant,” Philosophical Transactions of the Royal Society A 310 (1983): 303-310.

14 Kleban, interview with author, January 17, 2008.

15 Steven B. Giddings, “The Fate of Four Dimensions,” Physical Review D 68 (2003).

CHAPTER 12

1 Matthew Kleban (New York University), interview with author, March 4, 2008.

2 Henry Tye (Cornell University), e-mail letter to author, May 15, 2008.

3 Ben Freivogel (University of California, Berkeley), interview with author, February 4, 2008.

4 Alejandro Gangui, “Superconducting Cosmic Strings,” American Scientist 88 (May 2000).

5 Edward Witten, “Cosmic Superstrings,” Physics Letters B 153 (1985): 243-246.

6 Alexander Vilenkin (Tufts University), interview with author, November 23, 2004.

7 David F. Chernoff (Cornell University), personal communication, February 8, 2010.

8 Tye, interview with author, February 8, 2010.

9 Alexander Vilenkin, “Alexander Vilenkin Forecasts the Future,” New Scientist, November 18, 2006.

10 Igor Klebanov (Princeton University), interview with author, April 26, 2007.

11 Rachel Bean (Cornell University), interview with author, April 25, 2007.

12 Joe Polchinski (University of California, Santa Barbara), interview with author, April 23, 2007.

13 Gary Shiu (University of Wisconsin), e-mail letter to author, May 19, 2007.

14 Tye, interview with author, January 22, 2007.

15 Cliff Burgess (Perimeter Institute), interview with author, April 13, 2007.

16 Bret Underwood (University of Wisconsin), interview with author, April 13, 2007.

17 Liam McAllister (Princeton University), interview with author, January 26, 2007.

18 Shiu, interview with author, May 19, 2007.

19 Sarah Shandera (Columbia University), interview with author, May 15, 2007.

20 Tye, interview with author, January 28, 2008.

21 Shiu, interview with author, May 19, 2007.

22 McAllister, interview with author, January 26, 2007.

23 Bean, interview with author, April 25, 2007.

24 Burt Ovrut (University of Pennsylvania), interview with author, February 2, 2007.

25 Underwood, interview with author, July 20, 2007.

26 Shiu, interview with author, July 20, 2007.

27 Polchinski, interview with author, August 31, 2007.

28 The Eöt-Washington Group, Laboratory Tests of Gravitational and Sub-Gravitational Physics, “Short-Range Tests of Newton’s Inverse-Square Law,” 2008-2009, http://www.npl.washington.edu/eotwash/experiments/shortRange/sr.html.

29 Shamit Kachru (Stanford University), interview with author, February 1, 2007.

30 K. C. Cole, “A Theory of Everything,” New York Times Magazine, October 18, 1987.

CHAPTER 13

1 Peter Goddard, ed., Paul Dirac: The Man and His Work (New York: Cambridge University Press, 1998).

2 Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics 13 (February 1960).

3 Chen Ning Yang, S. S. Chern: A Great Geometer of the 20th Century (Boston: International Press, 1998), p. 66.

4 Robert Osserman, Poetry of the Universe (New York: Anchor Books, 1996), pp. 142-143.

5 Richard P. Feynman, The Character of Physical Law (New York: Modern Library, 1994), p. 50.

6 Michael Atiyah, quoted in Patricia Schwarz, “Sir Michael Atiyah on Math, Physics and Fun,” The Official String Theory Web site, http://www.superstringtheory.com/people/atiyah.html.

7 Jim Holt, “Unstrung,” New Yorker, October 2, 2006, p. 86.

8 Michael Atiyah, “Pulling the Strings,” Nature 438 (December 22-29, 2005): 1081.

9 Robert Mills, “Beauty and Truth,” in Chen Ning Yang : A Great Physicist of the 20th Century, ed. Shing-Tung Yau and C. S. Liu (Boston: International Press, 1995), p. 199.

10 Henry Tye (Cornell University), e-mail letter to author, December 19, 2008.

11 Brian Greene, interview by Ira Flatow, “Big Questions in Cosmology,” Science Friday, NPR, April 3, 2009.

12 K. C. Cole, “A Theory of Everything,” New York Times Magazine, October 18, 1987.

13 Nicolai Reshetikhin (University of California, Berkeley), interview with author, June 5, 2008.

14 Robbert Dijkgraaf (University of Amsterdam), interview with author, February 8, 2007.

15 Brian Greene, The Elegant Universe (New York: Vintage Books, 2000), p. 210.

16 Andrew Strominger (Harvard University), interview with author, August 1, 2007.

17 S. Ramanujan, “On Certain Arithmetic Functions,” Transactions of the Cambridge Philosophical Society 22 (1916): 159-184.

18 Lothar Goettsche, “A Conjectural Generating Function for Numbers of Curves on Surfaces,” November 11, 1997, arXiv.org, Cornell University archives, http://arxiv.org/PS_cache/alg-geom/pdf/9711/9711012v1.pdf.

19 Ai-Ko Liu, “Family Blowup Formula, Admissible Graphs and the Enumeration of Singular Curves, I,” Journal of Differential Geometry 56 (2000): 381-579.

20 Bong Lian (Brandeis University), interview with author, December 12, 2007.

21 Michael Atiyah, “Pulling the Strings,” Nature 438 (December 22-29, 2005): 1082.

22 Glennda Chui, “Wisecracks Fly When Brian Greene and Lawrence Krauss Tangle over String Theory,” Symmetry 4 (May 2007): 17-21.

23 Sean Carroll, “String Theory: Not Dead Yet,” Cosmic Variance blog, Discover online magazine, May 24, 2007, http://cosmicvariance.com.

24 Edward Witten, quoted in K. C. Cole, “A Theory of Everything,” New York Times Magazine, October 18, 1987.

25 Ibid.

26 Alan Guth (MIT), interview with author, September 13, 2007.

27 Greene, The Elegant Universe (New York: Vintage Books, 2000), p. 261.

28 Max Tegmark (MIT), interview with author, October 23, 2007.

29 Faye Flam, “Getting Comfortable in Four Dimensions,” Science 266 (December 9, 1994): 1640.

CHAPTER 14

1 Cumrun Vafa (Harvard University), interview with author, January 19, 2007.

2 Robbert Dijkgraaf (University of Amsterdam), interview with author, February 8, 2007.

3 David Morrison (University of California, Santa Barbara), interview with author, May 27, 2008.

4 Allan Adams (MIT), interview with author, May 23, 2008.

5 Joe Polchinski (University of California, Santa Barbara), interview with author, August 31, 2007.

6 Edward Witten (Institute for Advanced Study), e-mail letter to author, January 30, 2007.

7 Adams, interview with author, May 23, 2008.

8 Turnbull WWW Server, “Quotations by Gauss,” School of Mathematical Sciences, University of St. Andrews, St. Andrews, Fife, Scotland, February 2006, http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Gauss.html.

9 Adams, interview with author, May 23, 2008.

10 Brian Greene, “String Theory on Calabi-Yau Manifolds,” lectures given at Theoretical Advanced Study Institute, 1996 session (TASI-96), Boulder, CO, June 1996.

11 K. C. Cole, “Time, Space Obsolete in New View of Universe,” Los Angeles Times, November 16, 1999.

12 Andrew Strominger (Harvard University), interview with author, August 1, 2007.

13 Morrison, interview with author, May 29, 2008.

14 Brian Greene, Elegant Universe (New York: Vintage Books, 2000), pp. 268, 273.

15 Paul Aspinwall (Duke University), interview with author, June 6, 2008.

16 Morrison, interview with author, May 27, 2008.

EPILOGUE

1 Andrew Strominger (Harvard University), interview with author, February 7, 2007.

2 Ibid.

3 Chris Beasley, Jonathan Heckman, and Cumrun Vafa, “GUTs and Exceptional Branes in F-Theory—I,” November 18, 2008, http://arxiv.org/abs/0802.3391; Chris Beasley, Jonathan Heckman, and Cumrun Vafa, “GUTs and Exceptional Branes in F-Theory—II: Experimental Predictions,” June 12, 2008, http://arxiv.org/abs/arxiv:0806.0102; Ron Donagi and Martijn Wijnholt, “Model Building with F-Theory,” March 3, 2008, http://lanl.arxiv.org/pdf/0802.2969v2; and Ron Donagi and Martijn Wijnholt, “Breaking GUT Groups in F-Theory,” August 17, 2008, http://lanl.arxiv.org/pdf/0808.2223v1.

4 Strominger, interview with author, July 23, 2007.

5 Cumrun Vafa (Harvard University), interview with author, November 2, 2007.

6 Leonard Mlodinow, Euclid’s Mirror (New York: Simon & Schuster, 2002), p. 255.

7 Edward Witten (Institute for Advanced Study), e-mail letter to author, February 12, 2007.

8 David Morrison (University of California, Santa Barbara), interview with author, May 27, 2008.

9 Ravi Vakil (Stanford University), interview with author, May 28, 2008.

10 Strominger, interview with author, August 1, 2007.

POSTLUDE

1 Piers Bursill-Hall, “Why Do We Study Geometry? Answers Through the Ages,” lecture, Faulkes Institute for Geometry, University of Cambridge, May 1, 2002.

2 Donald Zeyl (University of Rhode Island), interview with author, October 18, 2007.