Abstract
“Who among us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the advances of our science and at the secrets of its development in the centuries to come? What particular goals will the leading mathematical spirits of coming generations strive to reach? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?” Those words were spoken by the German mathematician David Hilbert on the morning of 8 August 1900 at the Sorbonne in Paris, in front of an audience comprising many of the most famous mathematicians in the world. David Hilbert taught at the University of Gottingen, and his fame had already been enshrined by the most influential scientists of the era. He was an elegant gentleman, balding, and with a well-groomed beard. His Panama hat and glasses completed a look that was at the same time serious and urbane.
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- 1.
Bernstein showed that solutions were analytical (i.e., most regular) if the minimum was of Class C3 (that is, with continuous derivatives of at least the third order). However, the existence of a minimum was guaranteed on a wider class of regularity; thus, no one could rule out that there could be solutions of a lower regularity class than C3. Successive studies extended Bernstein’s results, starting from a minimum with lower regularity.
- 2.
On the other side of the world De Giorgi was following the same events with apprehension, because his mother Stefania was born within the Austro-Hungarian Empire and had studied at a Hungarian school. Indeed, at that time Stefania was busy helping Hungarian refugees who had arrived as far as Lecce and who had difficulties communicating as they could not speak Italian. R. De Giorgi Fiocco, 2007.
- 3.
Non-linear processes are quite common in nature and they present themselves in contexts such as meteorology and aerodynamics, such as when a hurricane forms, or when an aircraft crosses the sound barrier. In a separate context, De Giorgi illustrated this problem by saying: “In order to explain non-linearity we should make a reference to a physical phenomenon evidencing it. When we think of a phenomenon in which a particular system is slightly perturbed, we are used to the idea that the resulting perturbation is proportional to the forces that cause it. For instance, if I take a string under tension and hang a small weight on it, it will sag slightly and proportionally to the weight I hang on it. So, if I hang a 1-kg weight and it sags 1 cm, I can assume that a 2-kg weight will cause a sagging of 2 cm. What this shows, and it follows from what can be commonly observed, is linearity, at least within a certain range of applied forces. When those forces increase beyond a certain limit the effects are no longer linear and the relationships become quite complex, and most certainly not proportional. In the case of weights on a string, we initially see linearity, then we no longer do, and then the string breaks and the phenomenon changes completely.” E. De Giorgi, in a presentation at the round table Descartes e dopo Descartes: il metodo, la matematica e le scienze, Lecce, October 1987. Published in [2].
- 4.
However, L. Nirenberg, who was around 30 years old at the time, and who became De Giorgi’s friend, does not remember this first meeting between Nash and De Giorgi. L. Nirenberg, email, 13 December 2006.
- 5.
S. Nasar, A Beautiful Mind, 1998.
- 6.
L. Carbone, 20 December 2007.
- 7.
R. Serapioni, Milan, 7 February 2009.
- 8.
S. Nasar, A Beautiful Mind, 1998.
- 9.
L. C. Piccinini comments (February 2007): “It seemed that Nash was in a big hurry to finish, as a phrase written between the lines of his article seems to show: ‘P.R. Garabedian writes from London of a manuscript by Ennio de Giorgi containing such a result.’ This phrase is not in evidence as much as it should have been. It is strange”. Garabedian’s mission is described in Chap. 4.
- 10.
E. De Giorgi never complained that he did not receive the prize. “He seemed serenely above these things,” commented his friend Giovanni Prodi (Pisa, October 2006).
- 11.
Dave Bayer, mathematician at Columbia University’s Barnard College, says that the publication ended up there by chance. D. Bayer, email, 19 September 2008.
- 12.
S. Nasar, A Beautiful Mind (1998).
- 13.
L. Nirenberg, email, 14 January 2007. Rota’s comments, which were included in Nasar’s book, were similar to what was being said in Pisa in the 1990s. However, this simple explanation seems improbable, because John Nash certainly had a predisposition for schizophrenia, and vicissitudes in what were frenetic times for him may have been contributing triggers. “I was not aware of this rumor,” stated Dave Bayer. D. Bayer, email, 19 September 2009.
- 14.
On this subject, De Giorgi said that, based on his experience, in his opinion a theorem is something that is discovered (because it is as if it already existed, no matter whether it is proven or not), whereas a proof is invented (because different mathematicians can find different proofs of a theorem). M. Emmer, Intervista con Ennio De Giorgi, Pisa, July 1996.
- 15.
E. Magenes in [7].
- 16.
M. Miranda, in La Matematica, Vol. 1, Einaudi (2007).
- 17.
L. C. Piccinini, February 2007.
- 18.
G. Prodi, Pisa, October 2006.
- 19.
M. Miranda, in La Matematica, Vol. 1, Einaudi (2007).
- 20.
E. De Giorgi, Sull’ analiticità delle estremali degli integrali multipli, Atti Acc. Naz. Lincei Rendiconti Cl. Sci. Fis. Mat. Natur. (8), 20 (1956).
- 21.
M. Curzio, 30 November 2007.
- 22.
The title was: Alcune applicazioni al calcolo delle variazioni di una teoria della misura k-dimensionale. In the article, De Giorgi wrote: “The theory we are referencing was presented for the first time by Renato Caccioppoli, and has been discussed by me in several published, or soon to be published, papers […] It implies the possibility of applying direct methods of calculus of variations to large classes of problems (among which we have specific cases such as Plateau’s Problem and the isoperimetric properties of a sphere), to establish the existence of a maximum or a minimum. This theory is also useful for the study of differential properties of solutions of variational problems. In this respect, I will cite a result related to a problem submitted to me by Prof. Guido Stampacchia”. The problem in question was, in fact, Hilbert’s nineteenth problem. Atti V Congr. Umi (Pavia 6–9/10/55), Cremonese (Rome 1956).
- 23.
C. Sbordone, 30 November 2007.
- 24.
L. C. Piccinini, email, 21 January 2008.
- 25.
J. F. Nash, email, 12 December 2007. Nash adds: “But of course it is quite different to be studying diffusion on a Riemannian manifold rather than just in Euclidean space.”
- 26.
G. Prodi, Pisa, February 2007.
- 27.
N. Uraltseva, Milan, 7 February 2009.
- 28.
O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and quasilinear elliptic equations, Academic Press (1968).
- 29.
D. Stroock, email, 8 February 2009. In a subsequent instance, Moser applied his methods to Nash’s demonstration as well. “Nash is a true genius,” comments Stroock “but he needs gifted people like Moser before most of us can follow him.”
- 30.
L. Caffarelli, email, 13 February 2009.
References
Bassani, F., Marino, A., Sbordone, C. (eds.): Ennio De Giorgi (Anche la scienza ha bisogno di sognare). Edizioni Plus, Pisa (2001)
Dossier Ennio De Giorgi. In: Guerraggio, A. (ed.) Lettera Matematica Pristem, pp. 27–28. Springer (1998)
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Parlangeli, A. (2019). Two on the Summit. In: A Pure Soul. Springer, Cham. https://doi.org/10.1007/978-3-030-05303-1_5
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