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:-
In 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.Manin, in collaboration with Sergei I Gelfand, published Methods of homological algebra in Russian in 1988. M M Kapranov writes:-
This 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.In the following year the same two authors published the sequel 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:-
This 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.Manin, in collaboration with A A Panchishkin, published Introduction to number theory in Russian in 1990 (English translation 1995, with a second edition in 2005). Yurey A Drakokhrust writes that:-
... 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.Other books by Manin include 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:-
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 a 54-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 the 1960s 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.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).
Three issues of the Moscow Mathematical Journal where dedicated to Manin for his 65th 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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