Abstract
In Pisa, De Giorgi’s mathematics developed to maturity. The work he had undertaken up to that point in measure theory, further consolidating what were also Caccioppoli’s ideas, helped him to face new and more complex subjects. Those ideas had already begun circulating in the United States, where they had been considered by Wendell Fleming, Fred Almgren, and Herbert Federer. Now, the more difficult part was using those techniques to demonstrate the regularity of minimal surfaces. De Giorgi, who, unlike most other researchers, did not care about publishing, concentrated on his objectives. The result was that for 2 years at least he did not publish anything.
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Notes
- 1.
M. Miranda in La Matematica, Vol. 1, Einaudi (2007).
- 2.
A. Chiffi (11 November 2007).
- 3.
L. Ambrosio, Pisa (13 February 2007).
- 4.
E. Giusti in [7].
- 5.
A problem that “De Giorgi himself had connected to the existence or otherwise of minimal surface cones,” says E. Giusti in [7]. So, if they existed at all, the singular points of a minimal hypersurface had to have a precise characteristic: in its immediate surroundings, the hypersurface had to look like a hyper-cone (its apex being a singular point). The problem was tackled and solved over subsequent years, as explained in the rest of the chapter.
- 6.
E. Giusti (Florence, 5 February 2007).
- 7.
W. Fleming (email, 5 March 2007).
- 8.
Seminario della Scuola Normale Superiore di Pisa.
- 9.
W. Fleming (email, 6 March 2007). W. Fleming also remembers (seminar in memory of De Giorgi, Pisa, 20–23 October 1997): “In August of 1962, J. P. Cecconi organised a workshop at Genova University where I met De Giorgi for the first time. Together with other Italian mathematicians, E. R. Reifenberg (known as Peter by his friends, author’s note) who had recently written an important paper on Plateau’s problem (Acta Mathematica, 1960) was also there.”
- 10.
W. Fleming, seminar in memory of De Giorgi, Pisa, 20–23 October 1997. Published in [5].
- 11.
S. Spagnolo, Pisa, 6 February 2007.
- 12.
In another context, and in agreement with what Spagnolo remembers, Sofri stated: “When I was a freshman I tried to resist bullying by senior students, who were quite hard on me, physically. When I became a sophomore I stupidly continued the bullying tradition, although for a short time.” A. Cazzullo, I ragazzi che volevano fare la rivoluzione, Sperling&Kupfer (Milan, 2006).
- 13.
As to the choice of this subject, see the commemoration of Ennio De Giorgi published in Boll. Umi (1999).
- 14.
S. Spagnolo, Pisa, 6 February 2007.
- 15.
Around US$ 500 at the time.
- 16.
At that time in Italy, utilities were state owned.
- 17.
D. Triscari (2007). According to Triscari it was there in Stockholm that De Giorgi met with S. Sobolev and found a mistake in his reasoning.
- 18.
According to the proceedings there were 2107 ordinary members and 984 associates.
- 19.
D. Triscari (2007).
- 20.
M. Miranda, La Matematica, Vol. 1, Einaudi (2007).
- 21.
G. De Cecco, Lecce, 22 December 2006.
- 22.
G. B. Rizza (2007).
- 23.
“Lucky you that you can watch the sea—Ungaretti was said to have told De Giorgi—I have to stay inside.” R. De Giorgi Fiocco, Lecce, 9 September 2007.
- 24.
E. De Giorgi, Cristoforo Colombo, 25 February 1964.
- 25.
A. De Giorgi, Lecce 2007.
- 26.
W. Fleming, email, 20 March 2007.
- 27.
E. De Giorgi, in a letter to his family, wrote: “Our arrival in New York was delayed because of a strike by harbor tug operators that forced the ship to perform difficult mooring procedures” (Providence, 26 February 1964).
- 28.
E. De Giorgi, Providence, 26 February 1964.
- 29.
W. Fleming, seminar in memory of De Giorgi, Pisa, 20–23 October 1997. Fleming’s presentation continued thus: “De Giorgi’s method has the advantage that it does not make use of the difficult Besicovitch theorem.
- 30.
W. Fleming, email, 20 March 2007.
- 31.
E. De Giorgi, Providence, 26 February 1964.
- 32.
Jaures P. Cecconi was a professor of mathematics at Genoa University.
- 33.
U. D’Ambrosio adds (email, 7 December 2007): “I met De Giorgi for the first time during the 1961–1962 academic year when I was studying for my doctorate in Genoa. He immediately gave me the impression of being a very kind person and a very creative mathematician. I was very impressed by his generosity in helping and suggesting new ideas and research directions for all those who attended his seminars and spoke with him.”
- 34.
E. De Giorgi, Providence, 1 April 1964.
- 35.
E. De Giorgi, Providence, 26 March 1964.
- 36.
R. De Giorgi Fiocco remembers (Lecce, 21 December 2007): “We sent it through the Perugina chocolate shop in Lecce.”
- 37.
E. De Giorgi, Providence, 10 April1964.
- 38.
E. De Giorgi, Providence, 25 April 1964.
- 39.
David Gilbarg (1918–2001). His obituary (http://news.stanford.edu/news/2002/june19/gilbarg-619.html) states: “[Gilbarg] was one of the very early researchers to appreciate the mathematical development of non-linear elliptic theory in the late 1950s by such mathematicians as De Giorgi, Nash, Morrey, and the Russian school led by Ladyzhenskaya and Uraltseva. These developments would later play a central role in the rapid development of the field of geometric analysis. Gilbarg was considerably ahead of his time in seeing this trend very early on, and his enthusiasm in this direction provided important impetus in the development of the geometric analysis area in the Mathematics Department.”
- 40.
L. Nirenberg does not remember the meeting (email, 1 January 2007). However, there is an anecdote relating to the event. “It was said that, while De Giorgi and Nash were in his office, Nirenberg exclaimed: ‘Mathematics is in my room!’”—E. Pascali (Lecce, 4 December 2006).
- 41.
S. Nasar, A Beautiful Mind (1998).
- 42.
E. Vesentini, seminar in memory of De Giorgi, Pisa, 20–23 October 1997.
- 43.
De Giorgi himself was pleased with the outcome that day. On 9 May he wrote to his family: “The trip to Boston went well: as I told you it was close by (one hour by train); I left in the morning, I did my seminar at 4 pm and that evening I caught the 10:30 pm train for Providence. In Boston, other than the MIT professors who had invited me, I also met with Andreotti and Vesentini, who are there this year as visiting professors. So, I toured the city and had dinner with them. After dinner, we stopped over at Prof. Segal’s house until it was time to take the train to Providence: just as in New York and Providence I found a warm welcome and my seminar was received with interest.” De Giorgi’s presentations ended ahead of schedule, because of the terse way he gave them owing to the language difficulties. This caused some problems during the debate that followed as De Giorgi had problems in maintaining a dialogue.
- 44.
E. De Giorgi (Stanford, 4 June 1964).
- 45.
R. Osserman, email, 2 November 2007.
- 46.
At Stanford, there were teachers such as George Pólya (1887–1985), Gabor Szegö (1895–1985), Charles Loewner (1893–1968), Stefan Bergman (1895–1977), and Max Schiffer (1911–1997).
- 47.
E. De Giorgi wrote to his family: “I write to you from Berkeley, where, as I mentioned the last time, I held a conference yesterday afternoon; as usual I found a warm welcome and my conference was followed with great interest.” The letter is dated Stanford, 26 May 1964, but this is probably an error (a more plausible date is 26 June 1964).
- 48.
Ibid.
- 49.
Some ambiguities on this matter were settled by S. Spagnolo who confirms: “I am sure that De Giorgi went to the United States only once, in 1964.” S. Spagnolo, email, 27 November 2007.
References
De Giorgi, E.: In: De Cecco, G., Rosato, M.L. (eds.) Hanno detto di lui…. Edizioni del Grifo, Lecce University, Department of Mathematics, Lecce (2004)
Dossier Ennio De Giorgi. In: Guerraggio, A. (ed.) Lettera Matematica Pristem, pp. 27–28. Springer (1998)
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Parlangeli, A. (2019). Incredible Ideas. In: A Pure Soul. Springer, Cham. https://doi.org/10.1007/978-3-030-05303-1_9
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