Italian mathematician (1924–2018)
Giovanni Battista Rizza (7 February 1924 – 15 October 2018), officially known as Giambattista Rizza, 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, 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. In 1956 he was in Rome at the INdAM, having been awarded a scholarship for his early research activities. A year later, in 1957, he was elected "discepolo ricercatore" in the same institute. During the same year, he gave some lectures on topics belonging to the field of several complex variables, later included in the lecture notes . 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, scoring first out of the three finalists: a year later, in 1962, he became extraordinary professor, and then, in 1965, ordinary professor to the same chair. In 1979 he became ordinary professor of "Geometria superiore", holding that chair uninterruptedly until 1994: 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.
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.
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.
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".
In 1999 the University of Parma, where he worked for more than 35 years, awarded him the title of professor emeritus.
Rizza was an honorary member of the Balkan Society of Geometers and life member of the Tensor Society.
Personality traits.
Enzo Martinelli described Giovanni Battista Rizza as a passionate researcher with a "strong intellectual force", and his scientific work as rich of geometrical ideas, denoting his strong algorithmic ability. According to Martinelli, Rizza is also a skilled organizer: 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. 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. 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 F on a general complex algebra A,
formula_1
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*, isomorphic to a given complex algebra A: precisely, he proves the formula
formula_2
where
Theory of analytic functions of several complex variables.
 ;templatestyles src="Template:Blockquote/styles.css" / ; All'estensione, tutt'altro che banale, allo spazio "R"2"n" dei metodi di Martinelli per dimostrare la (3), è dedicata una Memoria [8] di Giovanni Battista Rizza, il quale, sempre nell'ipotesi "ρ"("x"1, "y"1..., "xn", "yn") ∈ "C"ω, 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.
Rizza published only three work in this field: in the first one, the highly remarkable memoir , he extends to pluriharmonic functions of 2"n" 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. Precisely, he proves the following formula
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where
formula_3
Formula (1) express a condition the normal derivative of the boundary value of a pluriharmonic function on domain with real analytic boundary must satisfy. 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, and also to establish an integro-differential equation boundary values of pluriharmonic functions must satisfy. Rizza's result motivated other works on the same topic by Gaetano Fichera, Paolo de Bartolomeis and Giuseppe Tomassini.
References.
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