**Yuri Ivanovich Manin**'s parents were Ivan Manin and Rebecca Miller. He received his bachelor's degree from Moscow State University in 1958 but, even before this, his first mathematics paper

*On cubic congruences to a prime modulus*(Russian) appeared in print. He continued to undertake research at the Steklov Mathematical Institute in Moscow advised by Igor Shafarevich and after the award of his doctorate in 1960 he was appointed as Principal Researcher at the Steklov Mathematical Institute. During this period he published

*Moduli of a field of algebraic functions*(Russian) (1959) and

*On cubic congruences to a prime modulus*(1960).

In 1963 Manin submitted his habilitation thesis to the Steklov Mathematical Institute and two years later, in 1965, he was appointed Professor of Algebra at Moscow State University. The International Congresses of Mathematicians was held from 16 August to 26 August 1966. Manin presented a paper *Rational surfaces and Galois cohomology* in which he gave a survey of results, mainly due to himself, on classifying rational algebraic surfaces. He remained at Moscow State University until the early 1990s when he spent a while visiting Columbia University (1991-92), then Massachusetts Institute of Technology (1992-93) before accepting the position on the Board of Directors of the Max Planck Institute, Bonn in 1993. He became Professor emeritus at the Max Planck Institute in 2005. Earlier he had, in addition, accepted a professorship at Northwestern University, Evanston, in the United States.

The breadth of Manin's contributions have been remarkable. He has written papers on: algebraic geometry including ones on the Mordell conjecture for function fields and a joint paper with V Iskovskikh on the counter-example to the Lüroth problem; number theory including ones about torsion points on elliptic curves, *p*-adic modular forms, and on rational points on Fano varieties; and differential equations and mathematical physics including ones on string theory and quantum groups. He has also written famous papers on formal groups, the arithmetic of rational surfaces, cubic hypersurfaces, noncommutative algebraic geometry, instanton vector bundles and mathematical logic.

Let us look now at some of the books that Manin has published. In 1980 Manin and A I Kostrikin published the textbook *Linear algebra and geometry* in Russian. The second Russian edition was translated into English in 1989. In 1984 Manin published *Linear algebra and geometry* in Russian, it was translated into English in 1997. Also in 1984 he published *Gauge fields and complex geometry* in Russian. Raymond O Wells, Jr writes:-

Manin, in collaboration with Sergei I Gelfand, publishedIn the last two decades two new ideas were introduced into mathematical physics. The first of these is the notion of twistor geometry and the other is that of supersymmetry. In this book we find a beautiful blend of developments stemming from these two ideas written by a master expositor who uses the language of algebraic geometry to synthesize and unify the fundamental ideas involved.

*Methods of homological algebra*in Russian in 1988. M M Kapranov writes:-

In the following year the same two authors published the sequelThis is long-awaited modern textbook on homological algebra. The exposition is centred around the notion of derived category, which is treated in full detail for the first time in the textbook literature. ... the book contains more than just an exposition of the subjects involved. It teaches the reader, via examples, how to "think homologically". The book has many "levels of understanding" and can be read, with much benefit, on each of them. Thus, it can be used by students just beginning to study homological algebra, as well as by specialists who will find there some points which have never been clarified in the literature.

*Homological algebra*which is mainly aimed at presenting applications of the subject within mathematics.

In 1989 Manin, with I Yu Kobzarev, published *Elementary particles* although the work was essentially completed seven years earlier. H Araki writes:-

Manin, in collaboration with A A Panchishkin, publishedThis book presents a kind of philosophical assessment along with an introductory description of developments and the present status of quantum field theory as a basis for the theory of elementary particles.

*Introduction to number theory*in Russian in 1990 (English translation 1995, with a second edition in 2005). Yurey A Drakokhrust writes that:-

Other books by Manin include... the authors describe the current state of various aspects of number theory from a single point of view. They avoid traditional classification, stressing general ideas and principles that have application in different areas of the theory.

*Cubic forms: algebra, geometry, arithmetic*(Russian) (1972),

*A course in mathematical logic*(1977),

*Computable and noncomputable*(Russian) (1980),

*Quantum groups and noncommutative geometry*(1988),

*Topics in noncommutative geometry*(1991),

*Frobenius manifolds, quantum cohomology, and moduli spaces*(1999).

In our list above we have not mentioned Manin's fascinating book *Mathematics and physics* (1981). To illustrate Manin's thinking about mathematics and physics we quote from the Foreword:-

Manin has received, and continues to receive, many hoours for his outstanding mathematicl contributions. These include the Moscow Mathematical Society Award (1963), the Lenin Prize (1967), the Brouwer Gold Medal (1987), the Frederic Esser Nemmers Prize (1994), the Rolf Schock Prize (1999), the King Faisal Prize for Mathematics (2002), and the Georg Cantor Medal of the German Mathematical Society (2002).The author wishes to show how mathematics associates new mental images with such physical abstractions; these images are almost tangible to the trained mind but are far removed from those that are given directly by life and physical experience. For example, a mathematician represents the motion of planets of the solar system by a flow line of an incompressible fluid in a54-dimensional phase space, whose volume is given by the Liouville measure .... The computational formalism of mathematics is a thought process that is externalised to such a degree that for a time it becomes alien and is turned into a technological process. A mathematical concept is formed when this thought process, temporarily removed from its human vessel, is transplanted back into a human mold. To think ... means to calculate with critical awareness. The 'mad idea' which will lie at the basis of a future fundamental physical theory will come from a realization that physical meaning has some mathematical form not previously associated with reality. From this point of view the problem of the 'mad idea' is the problem of choosing, not of generating, the right idea. One should not understand that too literally. In the1960s it was said(in a certain connection)that the most important discovery of recent years in physics was the complex numbers. The author has something like that in mind.

Three issues of the *Moscow Mathematical Journal* where dedicated to Manin for his 65^{th} birthday. In an introduction, the editors give an indication of the breadth of his mathematical contributions and also write about his interests outside mathematics:-

His interests beyond mathematics are also extremely varied. Manin has published research and expository papers in literature, linguistics, glotto-genesis, mythology, semiotics, physics, history of culture, and philosophy of science. The example he set for those around him was not that of a monomaniac mathematician, but of a deep scholar with wide interests, for whom penetration into the mystery of knowledge is much more important than professional success.

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

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