# Giovanni Battista Rizza

Giovanni Battista Rizza
Born 7 February 1924 (age 92)
Piazza Armerina
Residence Parma
Nationality Italian
Fields
Institutions
Alma mater Università degli Studi di Genova
Doctoral students see the teaching activity section
Known for
Notable awards
Spouse Lucilla Bassotti

Giovanni Battista Rizza (born 7 February 1924) (at the registry office Giambattista Rizza)[3] is an Italian mathematician, working in the fields of complex analysis of several variables and in differential geometry: he is known for his contribution to hypercomplex analysis, notably for extending Cauchy's integral theorem and Cauchy's integral formula to complex functions of a hypercomplex variable,[4] the theory of pluriharmonic functions and for the introduction of the now called Rizza manifolds.

## Biography

The International Symposium on Algebraic Geometry held in Rome in 1965. Enrico Bompiani talking to Giovanni Battista Rizza and Vittorio Dalla Volta.

Born in Piazza Armerina, the son of Giovanni and Angioletta Bocciarelli, he graduated from the Università degli Studi di Genova, earning his laurea degree in 1949 under the direction of Enzo Martinelli.[5] In 1956 he was in Rome at the INdAM, having been awarded a scholarship for his early research activities.[6][7] A year later, in 1957, he was elected "discepolo ricercatore"[8] in the same institute.[9] During the same year,[10] he gave some lectures on topics belonging to the field of several complex variables,[11] later included in the lecture notes (Severi 1958).[12] In Rome he also met Lucilla Bassotti, who eventually become his wife. In 1961, he won the competitive examination for the chair of "Geometria analitica con elementi di Geometria Proiettiva e Geometria Descrittiva con Disegno" of the University of Parma,[13] scoring first out of the three finalists:[14] a year later, in 1962, he became extraordinary professor,[15] and then, in 1965, ordinary professor to the same chair.[16] In 1979 he became ordinary professor of "Geometria superiore",[17] holding that chair uninterruptedly until 1994:[18] from 1994 up to his retirement in 1997, he was "professore fuori ruolo" in the same department of mathematics where he worked for more than 35 years.[19]

Apart from his research and teaching work, he was actively involved as a member of the editorial board of the "Rivista di Matematica della Università di Parma", and served also as the journal director from 1992 to 1997.[20]

### Honors

In 1954 he was awarded the Ottorino Pomini prize by the Unione Matematica Italiana, jointly with Gabriele Darbo: the judging commission was composed by Giovanni Sansone (as the president), Alessandro Terracini, Beniamino Segre, Giuseppe Scorza-Dragoni, Carlo Miranda, Mario Villa and Enzo Martinelli (as the secretary).[1]

In 1973 He was awarded the golden medal "Benemeriti della Scuola, della Cultura, dell'Arte" by the President of the Italian Republic,[2] as an acknowledgement his research and teaching and achievements as civil servant at the University of Parma.[21]

In 1995, to celebrate his 70th birthday, an international conference on differential geometry was organized in Parma: the proceedings were later published as a special issue of the "Rivista di Matematica della Università di Parma".[22]

In 1999 the University of Parma, where he worked for more than 35 years, awarded him the title of professor emeritus.[23]

He is currently an honorary member of the Balkan Society of Geometers and life member of the Tensor Society.[24]

### Personality traits

Enzo Martinelli describes Giovanni Battista Rizza as a passionate researcher with a "strong intellectual force",[25] and his scientific work as rich of geometrical ideas, denoting his strong algorithmic ability.[26] According to Martinelli, Rizza is also a skilled organizer:[27] his ability in organizational tasks is also acknowledged and praised by Schreiber (1973, p. 1), who also alludes the positive opinions of colleagues and students alike about his involvement in research, teaching and administrative duties at the mathematics department of the University of Parma.

## Work

### Research activity

Giovanni Battista Rizza at work in his home office, in 2003.

Giovanni Battista Rizza has authored 53 research papers and 30 other scientific works, including research announcements, short notes, surveys and reports: he also wrote didactic notes and papers on historical topics, including commemorations of other scientists.[28] His main fields of research are the theory of functions on algebras, the theory of functions of several complex variables, and differential geometry.

#### Theory of functions on algebras

The theory of functions on algebras, also referred to as hypercomplex analysis, is the study of functions whose domain is a subset of an algebra.[29] The first works of Giovanni Battista Rizza belong to this field of research, and he was awarded the Premio Ottorino Pomini for his contributions.[4]

His first main result is the extension of Cauchy's integral theorem to every monogenic function F on a general complex algebra A,[30]

${\displaystyle \int _{\Gamma _{1}}\mathrm {F} (\mathrm {X} )\mathrm {d} \mathrm {X} =0}$

where Γ1 is a 1-dimensional cycle homologous to zero, and also satisfying other technical conditions.

Few years later, he extended Cauchy's integral formula to every monogenic function F on a commutative normed real algebra A*,[31] isomorphic to a given complex algebra A:[32] precisely, he proves the formula

${\displaystyle \int _{\Gamma _{1}}{\frac {\mathrm {F} (\mathrm {X} )}{\mathrm {X} -\Xi }}\mathrm {d} \mathrm {X} =2\pi i\sum _{s=1}^{k}\mathrm {N} ^{(s)}u^{(s)}\mathrm {F} (\Xi )}$

where

• X ≡ x* ≡ x identifies indifferently a point in the complex algebra A or in its isomorphic real algebra A*,
• Γ1 is again a 1-dimensional cycle homologous to zero, and satisfying other technical conditions,
• N(s) is the winding number of the cycle Γ1 respect to the zero divisor locus for the considered algebra.

#### Theory of analytic functions of several complex variables

All'estensione, tutt'altro che banale, allo spazio R2n dei metodi di Martinelli per dimostrare la (3), è dedicata una Memoria [8] di Giovanni Battista Rizza, il quale, sempre nell'ipotesi ρ(x1y1,..., xnyn) ∈ Cω, perviene a stabilire la (3) per n qualsiasi. Anche questo lavoro, per quanto redatto in lingua inglese e pubblicato su una delle principali riviste matematiche, non ha nella letteratura attuale, la notorietà che meriterebbe.[33]

— Gaetano Fichera, (Fichera 1982a, p. 135).

Rizza published only three work in this field:[34] in the first one, the highly remarkable memoir (Rizza 1955),[35] he extends to pluriharmonic functions of 2n real variables, n > 2, the methods introduced by Enzo Martinelli in order to give new proof of a result of Luigi Amoroso for pluriharmonic functions of four real variables.[36] Precisely, he proves the following formula

${\displaystyle {\frac {\partial u}{\partial \nu }}={\frac {\vert \nabla \rho \vert ^{3}}{Q(\rho )}}Eu}$

(1)

where

${\displaystyle \partial \Omega =\{x\in \mathbb {R} ^{2n}|\rho (x)=0\},}$

Formula (1) express a condition the normal derivative of the boundary value of a pluriharmonic function on domain with real analytic boundary must satisfy.[37] It can be used to construct an integral representation for pluriharmonic functions on such kind of domains, by using the Green's formula for the Laplacian,[38] and also to establish an integro-differential equation boundary values of pluriharmonic functions must satisfy.[39] Rizza's result motivated other works on the same topic by Gaetano Fichera, Paolo de Bartolomeis and Giuseppe Tomassini.[40]

### Teaching activity

During his teaching career at the University of Parma, Giovanni Battista Rizza was the academic advisor of several students. A partial list of his students includes the following ones:

## Notes

1. ^ a b The detailed motivation for the award is reported in the Bollettino UMI 1954, pp. 477–478. The high scientific value of the works of the two young mathematicians induced the commission to ask the benefactors supporting the prize for a double award: their request was accepted.
2. ^ a b
3. ^ See the list of the recipients of the medal "Benemeriti della Scuola, della Cultura, dell'Arte" and the Decreto ministeriale 17 febbraio 1999 conferring him the title of "Professor Emeritus".
4. ^ a b According to the motivation for the award of the "Premio Ottorino Pomini", reported on the Bollettino UMI (1954, p. 477), "Sono particolarmente degni di nota i risultati sui teoremi integrali per le funzioni regolari, sulle estensioni della formula integrale di Cauchy alle funzioni monogene sulle algebre complesse dotate di modulo commutative e sul conseguente sviluppo della relativa teoria, ed infine sulla struttura delle algebre di Clifford" ("Particularly notable results are the ones on the integral theorems for regular functions, the ones on the extension of Cauchy integral formula to complex commutative algebras with modulus, and lastly the ones on the structure of Clifford algebras").
5. ^ According to Martinelli (1994, p. 1), he was his first doctoral student.
6. ^ He, Giuseppe Arcidiacono and Dario Del Pasqua, were awarded the scholarship without sustaining the "colloquio" ("colloquium" in English translation), an oral exam where the candidate was asked to answer questions posed by a scientific jury, according to Roghi (2005, p. 46) who reports also an excerpt of the motivation given by the commission for the awarding of the scholarship to Rizza:-"...perché trattasi di giovani di cui è nota l'attività scientifica...", i.e. (English translation): "...because they are young researchers whose scientific activity is known, ...").
7. ^ Roghi (2005, pp. 8,29,277) also states that the scientific commission of the institute in charge in 1956 was still the first one, formed on 23 November 1939: its members were Francesco Severi (the president), Luigi Fantappie, Giulio Krall, Enrico Bompiani and Mauro Picone.
8. ^ "Disciple researcher" (English translation) was the appellation of junior research scientists working at the INdAM. See (Roghi 2005) for further details.
9. ^ See (Roghi 2005, p. 50).
10. ^ See (Roghi 2005, p. 50) and Severi (1958, p. III)
11. ^ See (Rizza 1958).
12. ^ Roghi (2005, p. 50) also precisely reports the costs carried by the INdAM to fund this course.
13. ^ "Analytic geometry with elements of projective geometry and descriptive geometry with drawing" (English translation).
14. ^ See the announce on the Bollettino UMI (1962, p. 454).
15. ^ See (Venturini 1963, p. 15).
16. ^ See the 1965 Yearbook of the University of Parma, p. 207: the exact date of this career advancement is 16th January 1965.
17. ^ Literally "higher geometry": it is an Italian university course on advanced geometry topics.
18. ^
19. ^ See the 1995 Yearbook of the University of Parma, pp. 887 and 1036: the locution, literally meaning "out of role professor", identifies a nearly retired professor which is not in charge of any particular university course.
20. ^ According to the timeline of Editors in Chief of the "Rivista", as reported in the historical section of the journal web site.
21. ^ See (Schreiber 1973, p. 1).
22. ^ See (Donnini, Gigante & Mangione 1994). In the preface, the editors and members of the organizing committee briefly commemorate Franco Tricerri, former pupil of Rizza and speaker at the conference, who died in a plane crash in China few weeks before the proceedings of the conference were published (ibid., p. iii).
23. ^ According to Decreto ministeriale 17 febbraio 1999.
24. ^ See the list of members of the Balkan Society of Geometers (2011) and of the Tensor Society (2010).
25. ^ Martinelli (1994, p. 1) precisely characterizes Rizza's scientific work as developed with "...molta passione e forza intellettuale...", i.e. with (English translation) "...much passion and intellectual force...".
26. ^ Again according to Martinelli (1995, p. 2):-"Queste poche righe mi auguro siano servite a dimostrare che Rizza è un matematico ricco di idee geometriche e dotato di forti capacità algoritmiche.", i.e. (free English translation) "I hope those few lines have been of some help in demonstrating that Rizza is a mathematician rich of geometrical ideas and gifted with a strong algorithmic ability."
27. ^ See (Martinelli 1994, p. 2).
28. ^ See, for example, (Rizza 1984), (Rizza 1986) and (Rizza 2002).
29. ^ For more information see the survey article by Rizza (1973) and the references cited therein.
30. ^ See (Rizza 1950).
31. ^ See (Rizza 1952), (Rizza 1952a) and the survey (Rizza 1973).
32. ^ In the terminology of Rizza (1952, 1952a), the algebra A* is said to be the real image of (precisely, l'immagine reale di) A.
33. ^ (English translation): "To the far from trivial extension to the R2n space of Martinelli's methods in order to prove (3) a Memoir [8] of Giovanni Battista Rizza is devoted, who, again under the hypothesis that ρ(x1, y1,..., xn, yn) ∈ Cω, succeeds in proving (3) for every n. Even this work, despite being written in English and published in a major mathematical journal, has not, in the current literature, the notoriety it deserves".
34. ^ The work (Rizza 1954) is only a research announcement related to the (Rizza 1955), while (Rizza 1958) is set of course notes based on the same paper and on (Rizza 1957).
35. ^ According to Fichera (1982b, p. 24), who praises this work as "molto considerevole": see also his comments in (Fichera 1982a, p. 135).
36. ^ See (Fichera 1982a, p. 135), (Fichera 1982b, pp. 24–25) and (Martinelli 1941).
37. ^ See (Fichera 1982a, p. 135), (Fichera 1982b, pp. 24–25) and (Fuks 1963, p. 277, footnote 1).
38. ^ See (Fichera 1982a, p. 134), (Fichera 1982b, p. 33) and (Martinelli 1941, p. 162).
39. ^ It is the Amoroso integro-differential equation: see (Fichera 1982a, p. 134) and (Fichera 1982b, pp. 33).
40. ^ See the hystorical survey sections in (Fichera 1982b, p. 25) and the work (de Bartolomeis & Tomassini 1981, p. 33).