Abstract
This steely phrase was spoken in the winter of 1953 by the brightest star of Italian mathematics of the time, the genial rebel Renato Caccioppoli, in front of an arena full of colleagues who looked on in astonishment. Caccioppoli had come to Rome to hold a series of seminars on measure theory, the same subject studied by De Giorgi. On that day, Hall A in the basement of the Institute of Mathematics was full. Rake thin, with a ghostly face, Caccioppoli was at the podium and all were silent. His arguments were not easy to follow: “Caccioppoli leaped from one idea to another with great speed, often leaving logical gaps that the listeners had to fill themselves (but his reasoning was always correct)—remembers Fernando Bertolini, who was there—Only those who were most informed could follow him, and no one dared contradict him.”
“Nothing is wilder than a pure soul.” (“Non c’è nulla di più barbaro di uno spirito puro”)
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Notes
- 1.
F. Bertolini, Gorzano, 18 February 2007.
- 2.
A few books have been written about him, among others Mistero Napoletano by E. Rea (Einaudi 1995). A movie was also made, Morte di un matematico napoletano, directed by M. Martone, in 1992.
- 3.
S. Stampacchia (February 2008), widow of Guido Stampacchia (1922–1978).
- 4.
L. Carbone, Naples, October 2006. M. Breiner adds (email, 21 January 2009): “During the referendum in 1946, when Italians were asked if they wanted to be a monarchy or a republic, Renato Caccioppoli was a very active republican. He had some apocryphal posters printed, in the name of Umberto II (the king at the time), and he had Guido Stampacchia and a few other friends affix them during the night in some of the less salubrious districts of Naples. The posters said something along the lines of ‘If the Monarchy wins the referendum, it promises to clean up Naples from all contraband, prostitution, and drug activities….’” This is a program that those people in Naples, at that time, would never have supported.
- 5.
G. Cimmino, Ricordo di Renato Caccioppoli, maestro e amico, Pisa, 10 April 1987. Published in [4].
- 6.
Episode recalled by F. Bertolini, Gorzano, 18 February 2007.
- 7.
E. Vesentini, commemoration, Pisa, 27 October 1996. Published in [5]. Bertolini remembers that De Giorgi’s intervention was stimulated by Picone, and that Caccioppoli was pleased about it.
- 8.
Renato Caccioppoli a 100 anni dalla nascita, Pristem/storia 8–9 (2004). André Gide was a French writer who won the Nobel Prize for Literature in 1947.
- 9.
E. Vesentini, commemoration, Pisa, 27 October 1996.
- 10.
E. De Giorgi, Su alcuni indirizzi di ricerca nel calcolo delle variazioni, Rome, 6–9 May 1985. De Giorgi’s text continues in this way: “So, [Picone] invited me to Naples for a few days where, by speaking directly with Caccioppoli, Miranda, and Stampacchia, I could experience first-hand the breadth of their ideas. This was much better than just reading their most interesting works. In their words, Caccioppoli, Miranda and Stampacchia blended their personal experience with those of their teachers Picone and Tonelli. In this way, the purpose of the direct method of the calculus of variations became clear, and so did the procedures linked to the four fundamental objectives: relaxation, semicontinuity theorems, representation theorems and regularity theorems.”
- 11.
R. De Giorgi Fiocco, 2007.
- 12.
M. Curzio, 30 September 2007.
- 13.
E. De Giorgi, Sviluppi dell’analisi funzionale nel Novecento, in the conference Il pensiero matematico del XX secolo e l’opera di Renato Caccioppoli, Pisa, 10 April 1987. From the archives of the Italian Institute for Philosophical Studies (1989).
- 14.
F. Bertolini, Gorzano, 18 April 2007.
- 15.
E. De Giorgi, L’artista dei numeri, L’Unità newspaper, 16 September 1992.
- 16.
Felix Hausdorff (1868–1942) was a German Jewish mathematician, who committed suicide with his wife and sister-in-law because of Nazi persecution.
- 17.
Constantin Carathéodory (1873–1950) was a Greek mathematician.
- 18.
E. De Giorgi, L’artista dei numeri, L’Unità newspaper, 16 September 1992.
- 19.
In a paper, De Giorgi himself wrote: “I had arrived at the same results at the same time and independently, starting from a different point of view and with different objectives. Caccioppoli proposed a general theory of integration of differential forms with more variables, and a complete extension of the Green–Stokes formulas. Instead, my initial objective was a substantial generalization of certain isoperimetric problems and I started a priori from the Gauss–Green formula.” E. De Giorgi, Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. 36 (1954).
- 20.
E. De Giorgi, L’artista dei numeri, L’Unità newspaper, 16 September 1992.
- 21.
The perimeter of a figure can be defined as the minimum limit of the perimeters of the polygons that approximate the area of the figure in question—E. De Giorgi, Sviluppi dell’analisi funzionale nel Novecento, in the conference Il pensiero matematico del XX secolo e l’opera di Renato Caccioppoli, Pisa, 10 April 1987.
- 22.
Both Caccioppoli and De Giorgi were interested in the particular case of co-dimension 1, i.e., to (n − 1)-dimensional hypersurfaces in a n-dimensional space, which is an extension of the concept of 2D surfaces in a 3D space.
- 23.
M. Miranda, Caccioppoli sets, Atti Acc. Naz. Lincei series 9 vol. 14 fasc. 3 (Rome 2003).
- 24.
E. De Giorgi, Definizione ed espressione analitica del perimetro di un insieme, Atti Acc. Naz. Lincei Rendiconti Cl. Sci. Fis. Mat. Natur. (1953).
- 25.
E. De Giorgi, Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. 36 (1954).
- 26.
The problem was originally tackled by Lagrange in the eighteenth century, then rediscovered and reformulated by, among others, the German Carl Friedrich Gauss (1777–1855), the British George Green (1793–1841) and the Ukrainian Mikhail Vasilyevich Ostrogradsky (1801–1862).
- 27.
V. L. Ambrosio in [3] and [6] and M. Miranda, Caccioppoli sets, Atti Acc. Naz. Lincei series 9 vol. 14 fasc. 3 (Rome 2003).
- 28.
E. De Giorgi, Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio a r dimensioni, Ric. Mat. 4 (1955).
- 29.
E. De Giorgi, Sviluppi dell’analisi funzionale nel Novecento, at the conference Il pensiero matematico del XX secolo e l’opera di Renato Caccioppoli, Pisa, 10 April 1987.
- 30.
“Caccioppoli had a reputation for publishing papers that were not entirely correct (although Caccioppoli’s intuition was very good),” remembers Fleming. He also admits that Young wasn’t very diligent in reading carefully the work of others, and that often it was hard to find people to review articles, as it was considered a low-priority job. W. Fleming, email, 6 March 2007.
- 31.
W. Fleming, email, 6 March 2007.
- 32.
“E. De Giorgi contributed the most to the revaluation of Caccioppoli’s work in Geometric Measure Theory.” G. Letta, Pisa, 6 February 2007.
- 33.
L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, S. Spagnolo, Ennio De Giorgi, Boll. Umi, Sect. B (8) 2 (1999).
- 34.
E. De Giorgi, Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. Atti Acc. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. 1 (8) 5 (1958). In this article, De Giorgi examined the isoperimetric problem of a hypersphere in an arbitrary space of n-dimensions and with co-dimension 1, that is, considering (n − 1)-dimensional hypersurfaces. Caccioppoli read this article, and, on 24 February 1958, wrote this letter to Picone: “Dear Mauro, forgive me for my usual delay […] As usual, the substance is good and the form not exactly brilliant. In a separate sheet I suggest the small necessary corrections that can be made to the sloppy manuscript, which doesn’t require much rewriting. Tweak the author’s ears and pass my best regards to your wife. Greetings, Renato.” Renato Caccioppoli a 100 anni dalla nascita, Pristem/storia 8–9 (2004).
- 35.
E. De Giorgi, Il calcolo delle variazioni: origini antiche e prospettive future, presentation in remembrance of L. Tonelli, Pisa, 12 March 1996. Published in L. Tonelli e la matematica nella cultura italiana del ’900, Sns (1998).
- 36.
W. Fleming, email, 6 March 2007.
- 37.
W. Fleming, email, 6 March 2007. Pauc and De Giorgi probably met in Italy, on the occasion of a Centro Internazionale Matematico Estivo (CIME) conference, with the title Quadratura delle superfici e questioni connesse, which was held at Villa Monastero, Varenna, 16–25 August 1954.
- 38.
De Giorgi’s results were always valid in co-dimension 1, whereas Federer and Fleming expanded these results to a set of arbitrary co-dimensions. L. Ambrosio in [3] and [6] explains: “Modern Geometric Measure Theory, which was probably born in 1969, the year of publication of Federer’s monograph, derives from a fortunate synthesis between the ideas developed by the Italian school (and particularly by Caccioppoli and De Giorgi) and the Theory of Rectifiable Sets and of integration based on measures of Carathéodory’s type.”
- 39.
E. De Giorgi, Un esempio di non unicità della soluzione di un problema di Cauchy, relativo ad un’equazione differenziale lineare di tipo parabolico, Rend. Mat. Appl. (5) 14 (1955).
- 40.
One was identically null, and the other was given explicitly. According to S. Spagnolo, “it was original the fact that Ennio built, one piece at a time, the equation and the solution”. S. Spagnolo, email, 26 September 2008. Spagnolo added that this was an interesting result, also because it concerned a problem of evolution. In classical physics, indeed, the evolution of a system is usually deterministic. This is based on the uniqueness of the solution of the evolution equations normally used in physics. De Giorgi’s counterexample contradicted these principles.
- 41.
S. Spagnolo, email, 26 September 2008.
- 42.
See Chap. 21.
- 43.
G. Prodi, Pisa, October 2006.
- 44.
Actually, Ladyzhenskaya does not appear in the list of participants, whereas De Giorgi does. Proceedings of the International Congress of Mathematicians Amsterdam 02-09/09/54. Erven P. Noordhoff NV Groningen and North-Holland Publishing Co Amsterdam North-Holland (1957). Maybe Ladyzhenskaya was there unofficially, without having registered formally, or maybe they met on another occasion. But either way, it is quite certain that the two met face to face, sometime in the 1950s, as N. Uralceva confirms (Milan, 7 February 2009). Giovanni Prodi and his wife remember De Giorgi stopping by their home in Milan, on his way to the Netherlands for the congress (Pisa, October 2006). On that occasion, his presentation lasted 15 min, at the end of Wednesday, 8 September, and was entitled: “Una nuova definizione di varietà k-dimensionale orientata e di misura k-dimensionale di un insieme di uno spazio r-dimensionale.”
- 45.
F. Bertolini, email, 7 January 2009. The title of the seminar was “Quadratura delle superfici e questioni connesse” (Villa Monastero, Varenna, 16–25 August 1954). There were 29 registered participants, 25 of whom were of Italian nationality.
- 46.
L. Carbone remembers (email, 8 January 2009), that every now and again, E. De Giorgi cited one of Pauc’s books, probably “Les méthodes directes en calcul des variations et en géometrie différentielle,” Hermann et Cie ed. (Paris 1941). In the same year, however, Pauc wrote another, shorter, book, also on the calculus of variations.
- 47.
Sergei Lvovich Sobolev (1908–1989) was a Soviet mathematician, who made important contributions to the field of functional analysis and to the theory of partial differential equations.
- 48.
A. Chiffi, 11 November 2007.
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Parlangeli, A. (2019). A Pure Soul. In: A Pure Soul. Springer, Cham. https://doi.org/10.1007/978-3-030-05303-1_3
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