**Hans Grauert**'s parents were Clemens and Maria Grauert. He was born in Haren-Ems which is in Niedersachsen (Lower Saxony) in the north of Germany close to the border with The Netherlands. He attended primary school, the local Volksschule, in Haren-Ems from the age of six. After five years, the last two of which were made difficult by World War II, he entered the Mittelschule in Haren-Ems in 1941. Hans' four years at this school were war years but when he left at the end of March 1945 the Allied armies had already crossed the Rhine and Germany was close to defeat. It would be almost a year before Grauert could continue his education.

The war had ended by the time Grauert began his studies at the Gymnasium in Meppen, a town about 12 km due south of Haren-Ems. He studied at the school from January 1946 until he graduated in February 1949. Following this he spent the summer semester at the University of Mainz before moving to the University of Münster where he began his studies in the autumn of 1949. He studied at Münster undertaking research towards his doctorate in 1952. After spending time at the Eidgenössische Technische Hochschule Zürich in 1953 where he was influenced by Beno Eckmann, he returned to Münster where he submitted his doctoral thesis *Charakterisierung der Holomorphiegebiete durch die vollständige Kählersche Metrik* Ⓣ. He was awarded his doctorate by the University of Münster on 30 July 1954. His first paper *Métrique kaehlérienne et domaines d'holomorphie* Ⓣ was published in French in the same year.

During the next year Grauert was able to undertake work supported by grants from Nordrhein-Westfalen and from Deutsche Forschungsgemeinschaft. He published three papers in 1955, namely *Charakterisierung der holomorph vollständigen komplexen Räume; Zur Theorie der Modifikationen. I.* Ⓣ * Stetige und eigentliche Modifikationen komplexer Räume* Ⓣ; and (with Reinhold Remmert) *Fonctions plurisousharmoniques dans des espaces analytiques. Généralisation d'une théorème d'Oka* Ⓣ. In September 1955 Grauert was appointed as an assistant at the University of Münster submitting his habititation thesis there in February 1957. He spent the year 1957-58 at the Institute for Advanced Study at Princeton, then the spring semester of 1959 at the Institut des Hautes études Scientifique in Paris. His output of published papers was quite remarkable with four papers being published in 1956 and six papers in 1957.

On 1 September 1959 Grauert was appointed as an ordinary professor at the University of Göttingen to fill the chair which Carl Ludwig Siegel had occupied. At Göttingen he supervised the doctoral studies of 44 students, see [3] for details, several of whom collaborated with him on major projects. In 1994 Grauert's selected papers were published in [2]. The papers are arranged under three headings which indicate the main areas of his research: General theory of complex spaces, Levi problem and pseudoconvexity, Fibre bundles, Direct images, *q*-convexity and cohomology, Deformation of complex objects, Decomposition of complex spaces, and Special results (which contains three papers on complex manifolds). Other areas on which Grauert wrote papers, but are not included in [2], are hyperbolicity, non-Archimedean function theory and quantum physics. John E Fornaess reviewing the two volume work writes:-

Hans Grauert has been the leading mathematician in the theory of several complex variables in his generation.

Let us look briefly at some books Grauert has written. Together with Ingo Lieb (one of his doctoral students who obtained his Ph.D. in 1965) he published *Differential- und Integralrechnung. I* Ⓣ in 1967. In the following year Grauert published *Differential- und Integralrechnung. II* Ⓣ, this second volume being written jointly with Wolfgang Fischer (another doctoral student of Grauert who obtained his Ph.D. in 1964). Also in 1968 the third volume *Differential- und Integralrechnung. III* Ⓣ was published, this one being a collaboration between Grauert and Lieb. Each volume ran to several editions, for example the fourth edition of the first volume appeared in 1976. In 1971 Grauert published *Analytische Stellenalgebren* Ⓣ written jointly with Reinhold Remmert. The book studies local algebras of complex analytic spaces. K Wolffhardt writes in a review:-

The book is written very carefully and precisely. Interesting examples are described in detail. The book is recommended to everyone interested in analytic algebras who has the[necessary]prerequisites...

In 1974 Grauert and Klaus Fritzsche (yet another doctoral student of Grauert who obtained his Ph.D. in 1975) published *Einführung in die Funktionentheorie mehrerer Veränderlicher* Ⓣ. Wells writes:-

This text is an excellent introduction to the classical themes of modern several complex variables theory: domains of holomorphy, holomorphic complexity, pseudoconvexity, the ring of convergent power series, analytic subvarieties and the several variables version of the Mittag-Leffler and Weierstrass problems ... The development of the basic theory is done very well, with pedagogically careful definitions, motivating examples and precise(sometimes even a little pedantic)proofs of various important results. ... The examples in the text are numerous and often excellent, and form a useful core of the text. ... Overall this book is highly recommended as an introduction to the subject of several complex variables.

In 1977 Grauert and Remmert published *Theorie der Steinschen Räume* Ⓣ (an English translation *Theory of Stein spaces* appeared in 1979 and was reprinted in 2004) and the same two authors published *Coherent analytic sheaves* in 1984. In 1999 Grauert and Hans-Christoph Grunau published *Lineare Algebra und analytische Geometrie* Ⓣ. In the book they explain its origin and Grauert's ideas about teaching are also brought out:-

This book evolved from a course given by the elder of the two authors in the winter semester1987/88at the University of Göttingen, prepared by the younger author and initially published by the Mathematical Institute of the University of Göttingen. ... Note that on one hand we place great importance on geometry, particularly on the interface between geometric visualization and mathematical-logical formulation; on the other hand, we also treat in this book a large part of the basic theory that is needed, say, by students of mathematical economics or physics, even though it only reflects the contents of the first part of the two-semester course 'Analytical geometry and linear algebra'. Much space is occupied by the treatment of systems of linear equations(the Gaussian algorithm), the theory of determinants and the theory of eigenvalues of linear mappings in 'Euclidean' vector spaces(transformation of principal axes). It is our goal to proceed quickly to the core of the topics mentioned and thus we deliberately dispense with the greatest possible generality. Numerous examples, which partly resemble solved exercises, are meant to help with the comprehension of new concepts or methods.

Finally we mention Grauert and Fritzsche's book *From holomorphic functions to complex manifolds* (2002) which is a graduate text designed, the authors write:-

... to give an understandable introduction to the theory of complex manifolds.

Grauert has received many honours for his outstanding contributions. He had received honorary doctorates from the universities of Bayreuth, Bochum, and Bonn. Among the academies who have elected him to membership we mention the German Academy of Scientists Leopoldina, the Göttingen Academy of Sciences, and the academies of Mainz, Catania and Munich. He was invited to give a 30 minute address to the International Congress of Mathematicians in Edinburgh in 1958, and a one hour address to the International Congress of Mathematicians in Stockholm in 1962. In 1991 the University of Erlangen awarded him the von Staudt Prize.

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