Giovanni Battista Rizza (7 February 1924 – 15 October 2018), officially known as Giambattista Rizza,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". was 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,According to the motivation for the award of the "Premio Ottorino Pomini", reported on the , "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"). the theory of pluriharmonic functions and for the introduction of the now called Rizza manifolds.
Biography
Life and academic career 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.According to , he was his first doctoral student. In 1956 he was in Rome at the INdAM, having been awarded a scholarship for his early research activities.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 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, ..."). 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. A year later, in 1957, he was elected "discepolo ricercatore""Disciple researcher" (English translation) was the appellation of junior research scientists working at the INdAM. See for further details. in the same institute.See . During the same year,See and he gave some lectures on topics belonging to the field of several complex variables,See . later included in the lecture notes . also precisely reports the costs carried by the INdAM to fund this course. 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,"Analytic geometry with elements of projective geometry and descriptive geometry with drawing" (English translation). scoring first out of the three finalists:See the announce on the . a year later, in 1962, he became extraordinary professor,See . and then, in 1965, ordinary professor to the same chair.See the 1965 Yearbook of the University of Parma, p. 207: the exact date of this career advancement is 16th January 1965. In 1979 he became ordinary professor of "Geometria superiore",Literally "higher geometry": it is an Italian university course on advanced geometry topics. holding that chair uninterruptedly until 1994:See the 1980 Yearbook of the University of Parma, p. 209. 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.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.
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.According to the timeline of Editors in Chief of the "Rivista", as reported in the historical section of the journal web site.
Rizza died in Parma on 15 October 2018, at the age of 94.
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).
In 1973 he was awarded the golden medal "Benemeriti della Scuola, della Cultura, dell'Arte" by the President of the Italian Republic, as an acknowledgement his research and teaching and achievements as civil servant at the University of Parma.See .
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".See . 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 (p. iii). In 1999 the University of Parma, where he worked for more than 35 years, awarded him the title of professor emeritus.According to Decreto ministeriale 17 febbraio 1999.
Rizza was an honorary member of the Balkan Society of Geometers and life member of the Tensor Society.See the list of members of the and of the .
Personality traits Enzo Martinelli described Giovanni Battista Rizza as a passionate researcher with a "strong intellectual force", precisely characterizes Rizza's scientific work as developed with "...molta passione e forza intellettuale...", i.e. with (English translation) "...much passion and intellectual force...". and his scientific work as rich of geometrical ideas, denoting his strong algorithmic ability.Again according to : "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." According to Martinelli, Rizza is also a skilled organizer:See . his ability in organizational tasks is also acknowledged and praised by , 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 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.See, for example, , and . His main fields of research were 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.For more information see the survey article by and the references cited therein. The first works of Giovanni Battista Rizza belong to this field of research, and he was awarded the Premio Ottorino Pomini for his contributions.
His first main result is the extension of Cauchy's integral theorem to every monogenic function on a general complex algebra ,See .
where 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 on a commutative normed real algebra ,See , and the survey . isomorphic to a given complex algebra :In the terminology of , the algebra is said to be the real image of (precisely, l'immagine reale di) . precisely, he proves the formula
where identifies indifferently a point in the complex algebra or in its isomorphic real algebra , is again a 1-dimensional cycle homologous to zero, and satisfying other technical conditions, is the winding number of the cycle respect to the zero divisor locus for the considered algebra.
Theory of analytic functions of several complex variables
Rizza published only three work in this field:The work is only a research announcement related to the , while is set of course notes based on the same paper and on . in the first one, the highly remarkable memoir ,According to , who praises this work as "molto considerevole": see also his comments in . he extends to pluriharmonic functions of real variables, , 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.See , and . Precisely, he proves the following formula
where is a polyharmonic function defined on a bounded domain , is a real analytic function defining the boundary of by the equation
is a linear combination of the Levi forms of relative to couples of complex variables, is a linear tangential operator defined on . Formula express a condition the normal derivative of the boundary value of a pluriharmonic function on domain with real analytic boundary must satisfy.See , and . 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,See , and . and also to establish an integro-differential equation boundary values of pluriharmonic functions must satisfy.It is the Amoroso integro-differential equation: see and . Rizza's result motivated other works on the same topic by Gaetano Fichera, Paolo de Bartolomeis and Giuseppe Tomassini.See the historical survey sections in and the work .
Selected publications
Research works . In this work Rizza extends the classical Cauchy's integral theorem to monogenic functions on a general complex algebra.
. A short research announcement describing briefly the results proved in . , available at DigiZeitschirften. . In this work Rizza epitomizes all known extensions of the Levi invariant to hypersurfaces in for in a single tensor of hybrid type. This paper is also interesting since it traces the story of such extensions back to the pioneering work of Eugenio Elia Levi. . The notes from the lectures given by Giovanni Battista Rizza for a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica: the full course notes, published as a monograph, include also a chapter by Enzo Martinelli and an appendix by Mario Benedicty). The topics he exposes are summarized by the two parts of the title, whose free English translations are "Explicit integral representation for –harmonic functions" and "Extension of the E. E. Levi invariant to the case of complex variables". . A short research announcement describing briefly the results proved in . . Another short presentation of the results proved in . . The article gives the proofs of the results previously announced in references and . . Shoshichi Kobayashi cites this article as the first one in the theory of Rizza manifolds. . .