Abstract
The atmosphere was tense in the Sala degli Stemmi, on the first floor of the Palazzo della Carovana, on that distant Tuesday, 3 March 1964. The speaker was Palmiro Togliatti, “The Best.” In front of him, challenging him with the arrogance of a 21-year-old, the face of a boy, was young Adriano Sofri.
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Notes
- 1.
Literally: Insignia’s Hall (translator’s note).
- 2.
Palmiro Togliatti (1893–1964) was until his death the undisputed leader of the Italian Communist Party. He participated at the Constituent Assembly (the parliamentary chamber that was formed in Italy just after the end of the Second World War from 25 June 1946 until 31 January 1948—translator’s note), and survived an assassination attempt on 14 July 1948. He was the brother of the mathematician Eugenio Togliatti (1890–1977).
- 3.
There were eight conferences that took place from December 1963 to March 1964.—P. Carlucci in L’archivio e la biblioteca come autobiografia, edited by L. Boccalatte, Franco Angeli (Milan, 2008).
- 4.
L. Radicati (Barbaricina, 9 February 2007). He did not remember De Giorgi being there on this occasion, even though he was usually present at these events and liked to speak.
- 5.
L. C. Piccinini (February 2007).
- 6.
Ibid.
- 7.
Lotta Continua (Continuous Struggle) was an extreme left wing extra-parliamentary political group formed in Italy in 1969 by students and workers at the Fiat car assembly plant in Turin (translator’s note).
- 8.
L. Radicati (Barbaricina, 9 February 2007).
- 9.
This is the expression that Radicati remembers (Barbaricina, 9 February 2007).
- 10.
This is the expression that G. Tomassini remembers (email, 11 January 2009).
- 11.
G. M. Cazzaniga (email, 2 February 2009).
- 12.
The Normale conference made the news, and was interpreted in different ways. Quazza was attacked by a right-wing Pisan magazine (Il Macchiavelli, 3 March 1964), whereas the weekly Gente of 18 June 1964 highlighted the extreme left leanings of the Scuola Normale (Source P. Carlucci in L’archivio e la biblioteca come autobiografia, edited by L. Boccalatte, Franco Angeli, Milan, 2008). On the other hand, immediately after the conference, when he returned to the editorial room of Rinascita, the weekly that he directed, Palmiro Togliatti was received as a hero for having stood up to the students (Source: A. Cazzullo, I ragazzi che volevano fare la rivoluzione, Sperling&Kupfer, Milan, 2006). Togliatti explicitly referenced the Pisan conference in an editorial in Rinascita on 14 March 1964.
- 13.
R. De Giorgi Fiocco (Lecce, 12 December 2007).
- 14.
E. Pascali (Lecce, 4 December 2006).
- 15.
A. Sofri in A. Cazzullo, I ragazzi che volevano fare la rivoluzione, Sperling&Kupfer (Milan, 2006).
- 16.
P. Carlucci in L’archivio e la biblioteca come autobiografia, edited by L. Boccalatte, Franco Angeli (Milan, 2008).
- 17.
P. Carlucci in La storia della Scuola Normale Superiore di Pisa in una prospettiva comparativa, edited by D. Menozzi and M. Rosa, Edizioni della Normale (Pisa, 2008).
- 18.
P. Carlucci in L’archivio e la biblioteca come autobiografia, edited by L. Boccalatte, Franco Angeli (Milan, 2008).
- 19.
Ibid.
- 20.
L. Radicati, Barbaricina, 9 February 2007. Often, Radicati and other professors of the time remember that the clashes within the Normale were limited to the appearance of protests, maybe just to seek attention: “One day, after contacting journalists, they left the school with their luggage—remembers Radicati—and they arranged to be photographed as they were leaving. But regret soon set in and the protest did not last long. They returned the next day without journalists.”
- 21.
M. Breiner, email, 28 January 2009.
- 22.
M. Breiner (2007).
- 23.
Centro Applicazioni Militari dell’Energia Nucleare (Nuclear Energy Military Applications Centre, translator’s note).
- 24.
M. Du Sautoy, The Music of Primes, Harper Collins (2003).
- 25.
E. Bombieri, One hundred reasons to be a scientist (Icpt 2004).
- 26.
E. Bombieri, email, 16 May 2007.
- 27.
Sergio Campanato (1930–2005) was an Italian mathematician.
- 28.
Not just Hölderian (Bombieri’s consideration). Technically, a real function f (defined over Rn) is Hölderian—in honour of the German mathematician Otto Hölder (1859–1937)—if there exist two real and non-negative constants C and α so that |f(x) − f(y)|≤ C|x − y|α. If α = 1, the function is said to be Lipschitzian in honour of the German mathematician Rudolph Lipschitz (1832–1903).
- 29.
E. Bombieri, E. De Giorgi e M. Miranda, Una maggiorazione a priori per le ipersuperfici minimali non parametriche, Arch. Rat. Mech. Anal. 32 (1969). It would appear that De Giorgi had announced this result a few months earlier at a national conference, but then preferred the new direction suggested by Bombieri for a demonstration.—E. Giusti, February 2007).
- 30.
James Harris Simons (1938–) is a unique personality. The son of a shoe manufacturer, he dedicated himself to mathematics when he was young and when he wrote a paper on minimal cones he was a researcher with the Institute for Defense Analyses in the USA. In 1982, he founded Renaissance Technologies Corp. in New York. In 2006 Forbes Magazine placed him at no. 278 of the richest men in the world, whereas the Financial Times classified him as the “most intelligent billionaire” (article by E. Lee and A. Katz, The alternative rich list, 22 September 2006).
- 31.
This demonstrated that De Giorgi kept up to date with the latest developments, but preferred not to show it. E. Bombieri, email, 16 May 2007.
- 32.
E. Bombieri, email, 16 May 2007.
- 33.
With this paper, the three mathematicians solved two distinct problems. The first concerned the regularity of minimal surfaces in general, that is, the argument that De Giorgi had been studying since the 1950s. Preceding papers by De Giorgi, Almgren, and Simons (in the 1960s) had demonstrated that, in a space of seven or less dimensions, these surfaces are perfectly regular. The new paper in 1968 showed that in eight-dimensional spaces, there could be minimal surfaces containing cone-shaped singularities (in a space of higher dimensions the singularities could be even more complex), defined by a relatively simple algebraic expression: \( {x}_1^2+{x}_2^2+{x}_3^2+{x}_4^2={x}_5^2+{x}_6^2+{x}_7^2+{x}_8^2 \), where x1, x2, x3, x4, x5, x6, x7, and x8 are the coordinates of the eight-dimensional space in question. The second problem consisted in building a hypersurface that could be described as the graph of a function defined on the entire space: it was Bernstein’s problem. E. De Giorgi himself had contributed towards the demonstration that, up until seven dimensions, the surfaces were in effect flat planes. In the eighth dimension, this was no longer true: Bombieri, De Giorgi, and Giusti demonstrated this by building an eight-dimensional hypersurface that was not a flat hyperplane.
- 34.
E. Giusti, Florence, 5 February 2007.
- 35.
E. Giusti, in “X-Day, I grandi della scienza del ’900: E. De Giorgi,” Quadro Film for RAI Ed. (2000). Text adapted.
- 36.
M. Forti, Lecce, 6 December 2006.
- 37.
E. Giusti (email, 19 December 2008).
- 38.
F. Bassani, Pisa, 8 February 2007. Bombieri himself confirms (email, 16 May 2007): “Many years after our collaboration, strolling past Pisa’s Duomo, De Giorgi said to me: ‘Enrico, working with you was a pleasure; but you see, mathematics must be enjoyed like a good bottle of wine, one sip at a time. We finished in two weeks, whereas we should have spent at least 2 years, one on the minimal cone and one on Bernstein’s problem.’ But he was clearly happy with the work we did.”
- 39.
M. Breiner (email, 27 December 2006). Breiner remembers that De Giorgi himself, when faced with a difficult problem, would say: “Here we need Bombieri’s techniques.”
- 40.
E. De Giorgi, Riflessioni sul senso della matematica, 27th conference of Scienza e Fede (Arliano, 8–9 June 1991). Published in [2].
- 41.
M. Breiner, email (2007).
- 42.
Other problems in the “drawer” were those on hyperbolic equations and Gevrey classes, and the problems of semi-continuity and relaxation. L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, S. Spagnolo, Ennio De Giorgi, Boll. Umi, Sect. B (8) 2 (1999).
- 43.
Ennio De Giorgi, Boll. Umi, Sect. B (8) 2 (1999).
- 44.
An analytical function is expressed point by point by a series (i.e., an infinite sum) of powers.
- 45.
L. Cattabriga and E. De Giorgi, Una dimostrazione diretta dell’esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti, Boll. Umi (4) 4 (1971). This paper encouraged the development of others: “One year later Lars Hörmander published the necessary and sufficient conditions. This completed the argument and De Giorgi never revisited the problem.” F. Colombini, Pisa, 12 February 2007.
- 46.
L. Nirenberg in [9].
- 47.
A. Marino, email, 11 November 2008.
- 48.
See Chap. 23.
References
Bassani, F., Marino, A., Sbordone, C. (eds.): Ennio De Giorgi (Anche la scienza ha bisogno di sognare). Edizioni Plus, Pisa (2001)
Ennio De Giorgi—Selected Papers. Springer (2006)
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Parlangeli, A. (2019). 1968. In: A Pure Soul. Springer, Cham. https://doi.org/10.1007/978-3-030-05303-1_11
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