**Aldo Andreotti**'s father was the noted Italian sculptor Libero Andreotti (1875-1933). After his secondary education, Aldo entered the Scuola Normale Superiore in Pisa in October 1942 where he attended courses by Leonida Tonelli. However, after divisions of the German army entered Italy in 1943, Andreotti escaped to Switzerland since all Italian men over the age of 18 risked being deported by the Germans to labour camps, from which many did not return. He made his way to Lausanne, on the shores of Lake Geneva, and attended courses at the university there, in particular attending Georges de Rham's lectures and those of Beno Eckmann. Andreotti was able to return to Italy after the end of World War II and completed his doctorate at Pisa in 1947 with a thesis on conformal representations.

Andreotti's research at Pisa resulted in five papers which were published in 1948: *Le serie lineari sopra retta multipla e in particulare sopra una retta doppia* Ⓣ; *Questioni di equivalenza relative alle curve riducibili e ai punti base di un fascio di curve sopra una superfice algebrica I* Ⓣ; *Questioni di equivalenza relative alle curve riducibili e ai punti base di un fascio di curve sopra una superfice algebrica II* Ⓣ; *Sulle corrispondenza fra due curve bi razionalmente distinte a moduli generali e sui modelli minimi dei loro prodotti* Ⓣ; and *Applicazione di un theorema di Schottley-Cecioni allo studio della geometria sopra una curve ellittica in relazione con quello sopra due curve ellittiche reali del tipo di Harnavk* Ⓣ. After graduating from Pisa, Andreotti went to Rome where he studied for three years with Francesco Severi. Severi was, by this time, involved in many non-mathematical pursuits, but these years, first as a research student at the National Institute of Advanced Mathematics (INdAM), and later as an assistant in geometry, were important years during which Andreotti became an extremely innovative mathematician. His interest in algebraic geometry, which had begun while he was in Pisa, developed greatly under Severi's guidance. Severi quickly realised that he had an extremely talented pupil and greatly admired Andreotti's talents.

In 1950 Andreotti went to the Princeton in the United States where he spent several months. This was an important period for Andreotti who came in contact with leading mathematicians such as Kunihiko Kodaira, Solomon Lefschetz, Carl Ludwig Siegel, Donald Spencer, André Weil and Oscar Zariski. This time at Princeton allowed Andreotti to greatly broaden his mathematical knowledge, for he learnt about the latest developments in algebra, general topology, algebraic topology and other areas which at that time were not being studied in Italy. He returned to Italy, entered the national competition for the selection of a full Professor of Geometry at the University of Turin and was interviewed in November 1951. Beniamino Segre was one of the judges on the selection committee that chose Andreotti who, within a few weeks, moved to Turin. He quickly made a substantial impact on the teaching of mathematics there [3]:-

Andreotti married Barbara Wilkinson Jenkins in 1953. Barbara was an American, the daughter of Professor John Jenkins and Ruth Parker from Athens, Georgia, USA. She had studied English literature and music, attending the University of Georgia, Radcliffe College and the University of Rochester. Before her marriage to Andreotti, she worked for the United States Government in Washington, D.C., and in Rome, Italy. Aldo and Barbara Andreotti had four children: Margherita, Maria Tecla, Libero and Felicita.In Turin, that little revolution in teaching began ... in1951, with the arrival of Aldo Andreotti. Then barely twenty-seven years old, Andreotti had reached an absolutely stunning maturity, coming in large measure from long months spent in the United States close to remarkable mathematicians ... At his arrival at Turin, Andreotti had some certainties. All Italian mathematicians were a bit ignorant, but especially the researchers of algebraic geometry; the old venerated master Severi was often an obstacle to updating because he didn't encourage young people enough to look at other schools which he considered competitive to the Italian one which he defended at all costs. It was absurd to continue to teach descriptive geometry in the first two years of the mathematics course and Andreotti replaced this by several topics in algebra such as ideal theory, polynomials, resultants etc.

In 1956 Andreotti left Turin and went to the University of Pisa. He spent the rest of his career at Pisa, although he spent much time abroad visiting the University of Nancy, the University of Paris, Göttingen University, the Institute for Advanced Study at Princeton, Strasbourg University, Brandeis University in Waltham, Massachusetts, Stanford University in Palo Alto, California and Oregon State University at Corvallis, Oregon. It was during one of his visits to the United States that he met Alexander Grothendieck who gives this description in [2]:-

Andreotti published over 100 papers. Up to 1958 his papers are all single-authored, but after that date the majority of them are co-authored with mathematicians such as Hans Grauert, Mauro Nacinovich, Raghavan Narasimhan, François Norguet, C Denson Hill, Wilhelm Stoll, and Edoardo Vesentini. His early work was on algebraic geometry but he went on to make major contributions to the theory of several complex variables and to partial differential operators. Perhaps his most famous results are his proof of the theorem of Leonida Tonelli (1958), his proof of the duality of Picard and Albanese varieties of algebraic surfaces, his work with A L Mayer on the Schottky problem (1967), and the Andreotti-Vesentini separation theorem which appeared in their joint 1965 paperAmong my many friends in this world ... the only other which I think has had a clear perception is Aldo Andreotti. I had made his acquaintance, and that of his wife Barbara and their twin children(even toddlers)in1955(at a party of André Weil's in Chicago, I think). ... Aldo had an acute sensitivity, which had in no way been dulled by dealings with mathematics and with "stars" like me. There was in him a gift of spontaneous sympathy for those he approached. This set him apart from all other friends I've known among mathematicians, or even outside mathematics. With him friendship always took precedence over common mathematical interests(which he did not lack), and he is one of the few mathematicians with whom I have talked about my life, and he about his. His father, like mine, was Jewish, and he had to suffer in Mussolini's Italy, like me in Hitler's Germany. I saw him always ready to encourage and support young researchers, in a climate where it became difficult to be accepted by the establishment. His interest was always spontaneously directed first to the person, not to "potential" mathematics or to fame. He was one of the most endearing people I have had the opportunity to meet.

*Carleman estimates for the Laplace-Beltrami equation on complex manifolds*. This theorem states that certain cohomology groups of coherent sheaves are separated. This result put into a differential-geometric setting finiteness theorems that Andreotti, in collaboration with Hans Grauert, had published three years earlier in

*Théorèmes de finitude pour la cohomology of the complexes espaces*Ⓣ concerning cohomology with coefficients on a locally free sheaf on a complex manifold. H Rossi, reviewing the 1965 paper, writes:-

Andreotti was invited to lecture at the International Congress of Mathematicians at Stockholm in 1962. He gave the addressThe authors give a lucid exposition of all the techniques used; in particular, there is a full discussion of the relevant topological spaces, called Fréchet-Schwartz spaces. Besides the finiteness theorems, there is also a general extension theorem for forms with coefficients in a vector bundle defined near the boundary of a complex manifold.

*Complex pseudoconcave spaces and automorphic functions*. He was invited again in 1970 when he gave the lecture

*E E Levi convexity and H Lewy problem*. Many of his invitations to give courses of lectures resulted in major publications such as

*Nine lectures on complex analysis*given at the Centro Internazionale Matematico Estivi in Bressanone, Italy in 1973 and

*Complexes of partial differential operators*which were the James K Whittemore Lectures in Mathematics he was invited to give at Yale University in May 1974. Monty Strauss, reviewing the monograph that was the published form of these Yale lectures writes:-

Other books by Andreotti are: (with Wilhelm Stoll)The author extends various classical results of the theory of Cauchy-Riemann equations to general complexes of linear partial differential operators. The monograph begins by examining elementary and Levi convexity and then discusses the Hans Lewy problem. Both the Dirichlet and Cauchy problems are examined in the context of cohomology theory through complexes of differential operators. Classical theorems on removable singularities and existence and uniqueness of solutions to the Cauchy problem are extended to some systems of partial differential operators through this technique.

*Analytic and algebraic dependence of meromorphic functions*(1971);

*Étude de géometrie algébrique*Ⓣ (1979); and (with Mauro Nacinovich)

*Analytic convexity and the principle of Phragmén-Lindelöf*(1980). A review of the first of these books begins as follows:-

One of Andreotti's students recalled a memorable lecture by Andreotti in a lecture room in the old Palazzo dei Cavalieri of the Scuola Normale Superiore:-The sophisticated expert may sneer at the inclusion of this and other introductory material and at the proofs of some facts "well known" to him and perhaps to him alone. As a plumber has to assemble and adjust his tools for a new job, so the mathematician has to ready his tools. Often enough, this task is neglected or omitted. Such a paper is difficult to read and may be even inaccessible to those less acquainted with the general subject area. Although this monograph is essentially a research paper, it is self contained and should be accessible to a reader who is familiar with the basic concepts of the field as provided in an introductory course or a standard textbook

Andreotti was awarded an honorary degree by the University of Nice. He was elected as a corresponding member of the Accademia Nazionale dei Lincei in 1968, awarded their Feltrinelli Prize in 1971, and became a full member in 1979. Almost certainly he would have received further awards but, sadly, he died at the young age of 55. He was buried in the church of San Francesco in Pisa. His friends and colleagues contributed the paper [7] as a tribute:-The students were waiting anxiously for the famous geometer, Aldo Andreotti, a legend of modern day Italian mathematics. ... a clumsy, chubby fellow shows up and announces that Professor Andreotti will be late, and that he will start the lecture in his stead. ... the fellow writes formulae monotonously, one after the other, on the wide chalky blackboard. ... suddenly, as if materializing out of thin air, Professor Andreotti shows up. He stares at the formulae, dismisses briskly the chubby and clumsy fellow, and begins writing, as if talking to himself. ... as suddenly as he had appeared, he stops. He sits on his chair, his jaws contracted, and looks at all of us, with a deep, almost annoyed, penetrating eye. There is a sense of oppression, of thick clouds gathering, a palpable mute tension. Professor Andreotti finally open his mouth and utters the following words, in a strong Tuscan accent(how strange it sounded in that momentous event): "Let me tell you one thing. Mathematics is hard, very hard. I worked all my life to understand it, and it is still hard. Each day. You must be prepared to work just as hard, or you better not waste your time, better go out and work in the fields". Then he pauses, an unfathomable pain engraved in his face, he toils with something on the desk, perhaps a cheap packet of cigarettes: "The lesson is over".

His wife, Barbara, died on 14 December 2008 in her home in Pisa, after a prolonged illness:-... we lost a good friend and a remarkable mathematician. For more than13years Andreotti has been a member of the Editorial Board of 'Compositio Matematica'. We shall remember his warm friendship, his fine personality and his mathematical ability.

She was known for her vast and international network of friends. An accomplished pianist, singer and teacher, avid traveller, lover of languages and literature, and generous friend, she leaves the cherished memory of a personality, sparkling with warmth and intelligence, and a life lived richly and fully, to all who knew and loved her.

**Article by:** *J J O'Connor* and *E F Robertson*

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