# Giovanni Battista Rizza

Giovanni Battista Rizza
Born February 7, 1924 (age 89)
Piazza Armerina
Residence Parma
Nationality Italian
Fields Several complex variables
Differential geometry
Institutions Università degli Studi di Genova
Istituto Nazionale di Alta Matematica
Sapienza University of Rome
Università degli Studi di Parma
Alma mater Università degli Studi di Genova
Doctoral students see the teaching activity section
Known for Integral representation of pluriharmonic functions
Rizza manifolds
Theory of functions on Algebras
Notable awards Premio Ottorino Pomini of the Unione Matematica Italiana (1954),[1]
Golden medal "Benemeriti della Scuola, della Cultura, dell'Arte" (awarded by the President of the Italian Republic) (1973)[2]
Spouse Lucilla Bassotti
Website
Giovanni Battista Rizza's web page at the website of the "Rivista di Matematica della Università di Parma".

Giovanni Battista Rizza (born February 7, 1924) is an Italian mathematician, working in complex analysis of several variables and in differential geometry: he is best known for his contribution to 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.[3] In 1956 he was in Rome at the INdAM, having been awarded a scholarship for his early research activities.[4][5] A year later, in 1957, he was elected "discepolo ricercatore"[6] in the same institute.[7] During the same year,[8] he gave some lectures on topics belonging to the field of several complex variables,[9] later included in the lecture notes (Severi 1958).[10] 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,[11] scoring first out of the three finalists:[12] a year later, in 1962, he become extraordinary professor,[13] and then, in 1965, ordinary professor to the same chair.[14] In 1979 he become ordinary professor of "Geometria superiore",[15] holding that chair uninterruptedly until 1994:[16] from 1994 up to his retirement in 1997, he was "professore fuori ruolo" in the same department of mathematics he worked for more than 35 years.[17]

### Honors

In 1954 he was awarded the Ottorino Pomini prize by the Unione Matematica Italiana, jointly with Gabriele Darbo: the judging commission included Giovanni Sansone (the president), Alessandro Terracini, Beniamino Segre, Giuseppe Scorza-Dragoni, Carlo Miranda, Mario Villa and Enzo Martinelli (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] acknowledging his research and teaching and achievements as civil servant at the University of Parma.[18]

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

He is currently an honorary member of the Balkan Society of Geometers and life member of the Tensor Society.[19]

### Tracts of his personality

Enzo Martinelli describes Giovanni Battista Rizza as a passionate researcher with strong intellectual force:[20] his scientific work is described a rich of geometrical ideas and denoting a strong algorithmic ability.[21] Martinelli praises also Rizza's skills as organizer:[22] his view is also shared by Schreiber (1973, p. 1), who refers the positive opinion of Rizza's colleagues and students about his involvement in research, teaching and administrative duties at the mathematics department of the University of Parma.

## Work

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 historical papers, including commemorations of other scientists.[23] His main fields of research are the theory of functions on algebras, the theory of functions of several complex variables, and differential geometry.

### Research activity

#### 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.[24] The first works of Giovanni Battista Rizza belong to this field of research, for which he was awarded the Premio Ottorino Pomini.[25] His first main result was the extension of the Cauchy integral formula to monogenic functions on the algebra $\scriptstyle A^\ast$:[26] precisely, he proved the following formula

$\int_{\Gamma_1}\frac{\mathrm{F}(\mathrm{X})}{\mathrm{X}-\Xi}\mathrm{d}\mathrm{X}=2\pi i\sum^k_{s=1}\mathrm{N}^{(s)}u^{(s)}\mathrm{F}(\Xi)$

## Selected publications

### Historical and commemorative papers

• Rizza, Giovanni Battista (1984), "Enzo Martinelli: Scienziato e Maestro", Rivista di Matematica della Università di Parma, (4) (in Italian), 10*: 1–10, MR 0777308, Zbl 0557.01011. "Enzo Martinelli: Scientist and Master" (English translation of the title) is a paper written by Rizza to honor his former master.
• Rizza, Giovanni Battista (Aprile 2002), "Commemorazione di Enzo Martinelli", Bollettino della Unione Matematica Italiana. Sezione A. La Matematica nella Società e nella Cultura, Serie VIII (in Italian) 5–A: 163–176, MR 1924344, Zbl 1194.01133. The "Commemoration of Enzo Martinelli" written by Rizza.

## Notes

1. ^ a b The detailed motivation for the awarding of the prize 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 approve a resolution to ask the benefactors supporting the prize for a double award, and they agreed.
2. ^ a b See the list of prize winners at the Presidenza della Repubblica Italiana (1973) web site.
3. ^ According to Martinelli (1994, p. 1), he was his first doctoral student.
4. ^ 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, ...").
5. ^ (Roghi 2005, p. 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.
6. ^ "Disciple researcher" (English translation) was the appellation of junior research scientists working at the INdAM. See (Roghi 2005) for further details.
7. ^ See (Roghi 2005, p. 50).
8. ^ See (Roghi 2005, p. 50) and Severi (1958, p. III)
9. ^ See (Rizza 1958).
10. ^ Roghi (2005, p. 50) also precisely reports the costs carried by the INdAM to fund this course.
11. ^ "Analytic geometry with elements of projective geometry and descriptive geometry with drawing" (English translation).
12. ^ See the announce on the Bollettino UMI (1962, p. 454).
13. ^ See (Venturini 1963, p. 15).
14. ^ See the Yearbook of the University of Parma (1965, p. 207): the exact date his advancement is the 16th January 1965.
15. ^ Literally "higher geometry": it is an Italian university course on advanced geometrical topics.
16. ^ See the Yearbook of the University of Parma (1980, p. 209).
17. ^ See the Yearbook of the University of Parma (1995, pp. 887 and 1036): the locution (literally "out of role professor") identifies a nearly retired professor which is not in charge of any particular university course.
18. ^ See (Schreiber 1973, p. 1).
19. ^ See the list of members of the Balkan Society of Geometers (2011) and of the Tensor Society (2010).
20. ^ Martinelli (1995, p. 1) precisely characterizes Rizza's scientific work as built with "...molta passione e forza intellettuale...", i.e. with (English translation) "...much passion and intellectual force...".
21. ^ Again according to Martinelli (1995, p. 2), who writes:-"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 words demonstrated that Rizza is a mathematician rich of geometrical ideas and gifted with a strong algorithmic ability."
22. ^ See (Martinelli 1994, p. 2).
23. ^ See for example references (Rizza 1984) and (Rizza 2002).
24. ^ For more information see the survey article by Rizza (1973) and the references cited therein.
25. ^ According to the published motivation for the award, 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" ("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").
26. ^ See (Rizza 1952), (Rizza 1952a) and the survey (Rizza 1973).