Теория струн и скрытые измерения Вселенной [заметки]

1

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

2

Так в книге. Правильно: «из двадцати треугольников» (прим. Vadi. Далее множество подобных «тупых» ошибок исправляю в тексте без примечаний)

3

Ibid., p. 46–47.

4

Ibid., p. 44.

5

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

6

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

7

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

8

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

9

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

10

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

11

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

12

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

13

Ibid.

14

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

15

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

16

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

17

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.

18

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

19

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

20

Mlodinow, Euclid's Window, p. 205.

21

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

22

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

23

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

24

Greene, The Elegant Universe, p. 72.

25

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

26

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.

27

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

28

Greene, interview with author, March 13, 2008.

29

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

30

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

31

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

32

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

33

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

34

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

35

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

36

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

37

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

38

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

39

Ibid.

40

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

41

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

42

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

43

Robert Greene, interview with author, April 17, 2008.

44

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

45

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

46

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

47

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

48

Ibid.

49

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

50

Calabi, interview, May 14, 2008.

51

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

52

Ibid.

53

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

54

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

55

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

56

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

57

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

58

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

59

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

60

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

61

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

62

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

63

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

64

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

65

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

66

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

67

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

68

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

69

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

70

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

71

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

72

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

73

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

74

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

75

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/0910/0910.5464vl.pdf.

76

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

77

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

78

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

79

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

80

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

81

Ibid.

82

Аллюзия на цитату из поэмы Томаса Стернза Элиота «Полые люди»: «Не взрыв, но всхлип». — Примеч. перев.

83

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

84

Ibid.

85

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

86

Plesser, interview with author, September 3, 2008.

87

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

88

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

89

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

90

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

91

Kachru, interview with author, August 19, 2008.

92

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

93

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

94

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

95

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

96

Ibid.

97

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

98

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

99

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

100

Plesser, interview with author, September 19, 2008.

101

Greene, interview with author, March 11, 2010.

102

Greene, The Elegant Universe, p. 259.

103

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

104

Greene, interview with author, March 13, 2010.

105

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

106

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

107

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

108

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

109

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

110

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

111

Gross, interview with author, September 19, 2008.

112

Ibid.

113

Gross, interview with author, September 24, 2008.

114

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

115

Gross, interview with author, September 24, 2008.

116

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

117

Strominger, interview with author, August 1, 2007.

118

Zaslow, interview with author, June 26, 2008.

119

Gross, interview with author, September 19, 2008.

120

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

121

Aspinwall, interview with author, June 23, 2008.

122

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

123

Aspinwall, interview with author, June 23, 2008.

124

Gross, interview with author, September 24, 2008.

125

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

126

American Mathematical Society, “Interview with Heisuke Hironaka,” Notices of the AMS 52, no. 9 (October 2005): 1,015.

127

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

128

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

129

Ibid.

130

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

131

Strominger, lecture.

132

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

133

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.

134

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

135

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

136

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

137

Ibid.

138

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

139

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

140

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

141

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

142

Ibid.

143

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/hepth/pdf/9711/9711053vl.pdf.

144

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

145

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

146

Cumrun Vafa (Harvard University), interviewwith author, September 26, 2008.

147

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

148

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

149

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

150

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

151

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

152

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

153

Taubes, “Black Holes and Beyond.”

154

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

155

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

156

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

157

Ibid.

158

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

159

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

160

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

161

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

162

Ovrut, interview with author, February 2, 2007.

163

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

164

Donagi, interview with author, November 19, 2008.

165

Candelas, interview with author, December 1, 2008.

166

Donagi, interview with author, May 3, 2008

167

Ovrut, interview with author, November 20, 2008.

168

Donagi, interview with author, November 20, 2008.

169

Ovrut, interview with author, November 20, 2008.

170

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

171

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

172

Candelas, interview with author, December 1, 2008.

173

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

174

Ovrut, interview with author, November 19, 2008.

175

Candelas, interview with author, December 1, 2008.

176

Strominger, interview with author, February 7, 2007.

177

Adrian Cho, “String Theory Gets Real — Sort Of,” Science 306 (November 26, 2004): 1,461.

178

Candelas, interview with author, December 1, 2008.

179

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

180

Gary Shiu, quoted in Adrian Cho, “String Theory Gets Real — Sort Of” Science 306 (November 26, 2004): 1,461.

181

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

182

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

183

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

184

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

185

Kachru, interview with author, September 8, 2007.

186

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

187

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

188

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

189

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

190

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

191

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

192

Mark Gross (Stanford University), interview with author, October 31, 2008.

193

Gross, interview with author, September 19, 2008.

194

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

195

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

196

Gross, interview with author, October 31, 2008.

197

Adams, interview with author, October 31, 2008.

198

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

199

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

200

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

201

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

202

Becker, interview with author, February 1, 2007.

203

Polchinski, interview with author, January 29, 2007.

204

Strominger, interview with author, August 1, 2007.

205

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

206

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

207

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

208

Sidney Coleman, “Fate of the False Vacuum: Semi-classical Theory,” Physical Review D. 15 (May 15, 1977): 2,929-2,936.

209

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

210

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

211

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

212

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

213

Giddings, interview with author, October 17, 2007.

214

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

215

Linde, interview with author, December 27, 2007.

216

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

217

Linde, interview with author, January 10, 2008.

218

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

219

Kleban, interview with author, January 17, 2008.

220

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

221

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

222

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

223

Sidney Coleman, “Fate of the False Vacuum: Semi-classical Theory,” Physical Review D. 15 (May 15, 1977): 2,929-2,936.

224

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

225

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

226

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

227

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

228

Giddings, interview with author, October 17, 2007.

229

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

230

Linde, interview with author, December 27, 2007.

231

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

232

Linde, interview with author, January 10,2008.

233

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

234

Kleban, interview with author, January 17, 2008.

235

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

236

Текст этого и последующих комментариев до конца Главы 12 отсутствует в издании. Похоже, что и комментарии с начала этой главы неверны (прим. Vadi)

237

То же.

238

То же.

239

То же.

240

То же.

241

То же.

242

То же.

243

То же.

244

То же.

245

То же.

246

То же.

247

То же.

248

То же.

249

То же.

250

То же.

251

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

252

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

253

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

254

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

255

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

256

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.

257

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

258

Michael Atiyah, “Pulling the Strings,” Nature 438 (December 22–29, 2005): 1,081.

259

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.

260

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

261

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

262

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

263

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

264

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

265

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

266

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

267

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

268

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

269

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

270

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

271

Michael Atiyah, “Pulling the Strings,” Nature 438 (December 22–29, 2005): 1,082.

272

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

273

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

274

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

275

Ibid.

276

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

277

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

278

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

279

Faye Flam, “Getting Comfortable in Four Dimensions,” Science 266 (December 9, 1994): 1, 640.

280

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

281

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

282

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

283

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

284

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

285

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

286

Adams, interview with author, May 23, 2008.

287

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

288

Ibid.

289

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

290

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

291

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

292

Morrison, interview with author, May 29, 2008.

293

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

294

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

295

Morrison, interview with author, May 27, 2008.

296

Andrew Strominger (Harvard University), February 7, 2007.

297

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/0808.2223vl.

298

Strominger, interview with author, July 23, 2007.

299

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

300

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

301

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

302

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

303

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

304

Strominger, interview with author, August 1, 2007.

305

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

306

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