## John W Milnor's awards

Putnam Fellow (1949), Putnam Fellow (1950), Fields Medal (1962), National Medal of Science (1967), Leroy P Steele Prize (1982), Wolf Prize (1989), Leroy P Steele Prize (2004), Leroy P Steele Prize (2011), Abel Prize (2011), Fellow of the American Mathematical Society (2014).

We now give some details and quotes from the citations of some of these awards. In a few cases we present extracts from Milnor's response.

**1. Putnam Fellow (1949).**

**2. Putnam Fellow (1950).**

**3. Fields Medal (1962).**

Proved that a7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology.

**4. National Medal of Science (1967).**

For clever and ingenious approaches in topology which have solved long outstanding problems and opened new exciting areas in this active branch of mathematics.

**5. Leroy P Steele Prize (1982).**

To John W Milnor for a paper of fundamental and lasting importance "On manifolds homeomorphic to the7-sphere, Annals of Mathematics64(1956),399-405".

**6. Wolf Prize (1989).**

The Wolf prize is awarded to John W Milnor for ingenious and highly original discoveries in geometry, which have opened important new vistas in topology from the algebraic, combinatorial, and differentiable viewpoint.

The highly original discoveries of Professor John W Milnor in geometry have exerted a major influence on the development of contemporary mathematics. The current state of the classification of topological, piecewise linear, and differentiable manifolds rests in large measure on his work in topology and algebra. Milnor's discovery of differentiable structures on S7^{}(the7-dimensional sphere)which are exotic, i.e. different from the standard structure, came as a complete surprise and marked the beginning of differential topology. Later, in joint work with Kervaire, Milnor turned these structures(on any S)^{n}into a group which could then(in part)be computed; it turns out that there are over sixteen million distinct differentiable structures on S31^{}![There are, in fact,16931177distinct differentiable structures on S31^{}.]In his important work in algebraic geometry on singular points of complex hypersurfaces, exotic spheres are related to links around singularities. In the combinatorial direction, Milnor disproved the long-standing conjecture of algebraic topology known as the 'Hauptvermutung', by constructing spaces with two polyhedral structures that cannot have a common subdivision. This was based on an unexpected use of the previously known concept of torsion, which has since become, in its various algebraic and geometric versions, a basic tool. These are just some highlights of Milnor's impressive body of work. Beyond the research papers, a wealth of new results are contained in his books. These are famous for their clarity and elegance and remain a source of continuing inspiration.

**7. Leroy P Steele Prize for Mathematical Exposition (2004).**

The Leroy P Steele Prize for Mathematical Exposition is awarded to John W Milnor in recognition of a lifetime of expository contributions ranging across a wide spectrum of disciplines including topology, symmetric bilinear forms, characteristic classes, Morse theory, game theory, algebraic K-theory, iterated rational maps É and the list goes on. The phrase "sublime elegance" is rarely associated with mathematical exposition, but it applies to all of Milnor's writings, whether they be research or expository. Reading his books, one is struck with the ease with which the subject is unfolding, and it only becomes apparent after reflection that this ease is the mark of a master. Improvement of Milnor's treatments often seems impossible. ... Milnor's many expository contributions to the mathematical literature have influenced more than one generation of mathematicians. Moreover, the examples that they provide have set a standard of clarity, elegance, and beauty for which every mathematician should strive.

**Extract from Milnor's Response.**

I have always suspected that the key to the most interesting exposition is the choice of a subject that the author doesn't understand too well. I have the unfortunate difficulty that it is almost impossible for me to understand a complicated argument unless I try to write it down. Over the years I have run into a great many difficult bits of mathematics, and thus I keep finding myself writing things down. ... I am very happy to report that as mathematics keeps growing, there are more and more subjects that I have to fight to understand.

**8. Leroy P Steele Prize for Lifetime Achievement (2011).**

The2011Steele Prize for Lifetime Achievement is awarded to John Willard Milnor. Milnor stands out from the list of great mathematicians in terms of his overall achievements and his influence on mathematics in general, both through his work and through his excellent books. His discovery of twenty-eight non-diffeomorphic smooth structures on the7-dimensional sphere and his further work developing the surgery techniques for manifolds shaped the development of differential topology beginning in the1950s. Another of his famous results from this period is a counterexample to the 'Hauptvermutung': an example of homeomorphic but not combinatorially equivalent complexes. This counterexample is a part of a general big picture of the relation between the topological, combinatorial, and smooth worlds developed by Milnor. Jointly with M Kervaire, Milnor proved the first results showing that the topology of4-dimensional manifolds is exceptional by revealing obstructions to the realization of2-dimensional spherical homology classes by smooth embedded2-spheres. This is one of the founding results of4-dimensional topology. In this way Milnor opened several fields: singularity theory, algebraic K-theory, and the theory of quadratic forms. Although he did not invent these subjects, his work gave them completely new points of view. For instance, his work on isolated singularities of complex hypersurfaces presented a great new topological framework for studying singularities and, at the same time, provided a rich new source of examples of manifolds with different extra structures. The concepts of Milnor fibers and Milnor number are today among the most important notions in the study of complex singularities. The significance of Milnor's work goes much beyond his own spectacular results. He wrote several books - Morse Theory(Princeton University Press, Princeton, NJ,1963), Lectures on the h-Cobordism Theorem(Princeton University Press, Princeton, NJ,1965), and Characteristic Classes(Princeton University Press, Princeton, NJ,1974), among others - that became classical, and several generations of mathematicians have grown up learning beautiful mathematical ideas from these excellent books. Milnor's survey "Whitehead torsion"(Bull. Amer. Math. Soc.72(3)(1966),358-426)provided an entry point for topologists to algebraic K-theory. This was followed by a number of Milnor's own important discoveries in algebraic K-theory and related areas: the congruence subgroup theorem, the computation of Whitehead groups, the introduction and study of the functor K2_{}and higher K-functors, numerous contributions to the classical subject of quadratic forms, and in particular his complete resolution of the theory of symmetric inner product spaces over a field of characteristic2, just to name a few. Milnor's introduction of the growth function for a finitely presented group and his theorem that the fundamental group of a negatively curved Riemannian manifold has exponential growth was the beginning of a spectacular development of the modern geometric group theory and eventually led to Gromov's hyperbolic group theory. During the past thirty years, Milnor has been playing a prominent role in development of low-dimensional dynamics, real and complex. His pioneering work with Thurston on the kneading theory for interval maps laid down the combinatorial foundation for the interval dynamics, putting it into the focus of intense research for decades. Milnor and Thurston's conjecture on the entropy monotonicity brought together real and complex dynamics in a deep way, prompting a firework of further advances. And, of course, his book Dynamics in One Complex Variable(Friedr. Vieweg & Sohn, Braunschweig,1999)immediately became the most popular gateway to this field. The Steele Prize honours John Willard Milnor for all of these achievements.

**Milnor's response.**

It is a particular pleasure to receive an award for what one enjoys doing anyway. I have been very lucky to have had so many years to explore and enjoy some of the many highways and byways of mathematics, and I want to thank the three institutions that have supported and inspired me for most of the past sixty years: Princeton University, where I learned to love mathematics; the Institute for Advanced Study for many years of uninterrupted research; and Stony Brook University, where I was able to reconnect with students and(to some extent)with teaching. I am very grateful to my many teachers, from Ralph Fox and Norman Steenrod long ago to Adrien Douady in more recent years; and I want to thank the family, friends, students, colleagues, and collaborators who have helped me over the years. Finally, my grateful thanks to the selection committee for this honour.

**9. Abel Prize (2011).**

All of Milnor's works display marks of great research: profound insights, vivid imagination, elements of surprise, and supreme beauty. Milnor's discovery of exotic smooth spheres in seven dimensions was completely unexpected. It signalled the arrival of differential topology and an explosion of work by a generation of brilliant mathematicians; this explosion has lasted for decades and changed the landscape of mathematics. With Michel Kervaire, Milnor went on to give a complete inventory of all the distinct differentiable structures on spheres of all dimensions; in particular they showed that the7-dimensional sphere carries exactly twenty-eight distinct differentiable structures. They were among the first to identify the special nature of4-dimensional manifolds, foreshadowing fundamental developments in topology. Milnor's disproof of the long-standing 'Hauptvermutung' overturned expectations about combinatorial topology dating back to Poincaré. Milnor also discovered homeomorphic smooth manifolds with non-isomorphic tangent bundles, for which he developed the theory of microbundles. In3-manifold theory, he proved an elegant unique factorization theorem. Outside topology, Milnor made significant contributions to differential geometry, algebra, and dynamical systems. In each area Milnor touched upon, his insights and approaches have had a profound impact on subsequent developments. His monograph on isolated hypersurface singularities is considered the single most influential work in singularity theory; it gave us the Milnor number and the Milnor fibration. Topologists started to actively use Hopf algebras and coalgebras after the definitive work by Milnor and J C Moore. Milnor himself came up with new insights into the structure of the Steenrod algebra(of cohomology operations)using the theory of Hopf algebras. In algebraic K-theory, Milnor introduced the degree2functor; his celebrated conjecture about the functor - eventually proved by Voevodsky - spurred new directions in the study of motives in algebraic geometry. Milnor's introduction of the growth invariant of a group linked combinatorial group theory to geometry, prefiguring Gromov's theory of hyperbolic groups. More recently, John Milnor turned his attention to dynamical systems in low dimensions. With Thurston, he pioneered "kneading theory" for interval maps, laying down the combinatorial foundations of interval dynamics, creating a focus of intense research for three decades. The Milnor-Thurston conjecture on entropy monotonicity prompted efforts to fully understand dynamics in the real quadratic family, bridging real and complex dynamics in a deep way and triggering exciting advances. Milnor is a wonderfully gifted expositor of sophisticated mathematics. He has often tackled difficult, cutting-edge subjects for which no account in book form existed. Adding novel insights, he produced a stream of timely yet lasting works of masterly lucidity. Like an inspired musical composer who is also a charismatic performer, John Milnor is both a discoverer and an expositor.

**Milnor's Response.**

The field of mathematics is a marvellous mosaic built up out of contributions by people from many different cultures, speaking many different languages, and stretching back over many hundreds of years. From the beginning mathematics has had a dual nature, partly abstract and self contained, but intimately concerned with understanding of the physical world around us. Much important mathematics was first inspired by real world problems; and mathematics has often contributed in totally unexpected ways. No one could have guessed that Riemann's study of curvature would form the basis for Einstein's theory of gravity; or that Hilbert's theory of infinite dimensional vector spaces would provide the foundations for quantum mechanics. The British mathematician G H Hardy proudly bragged that his work in number theory would never be sullied by applications. He would have been horrified to learn that it is now the basis for methods in cryptography which are fundamentally important in commercial applications and also in military applications. The connections between mathematics and other sciences work in both directions. Claude Shannon's work on communication theory was inspired by the work of physicists on statistical mechanics, and now has important applications not only in computer science but also in the mathematical theory of dynamical systems. The mathematical theory of Riemannn surfaces is now very important to mathematical physicists. Conversely, tools developed by mathematical physicists play a very important role in topology. I have been very lucky to have been able to enjoy this magnificent mosaic of mathematics for more than sixty years, and to make some contributions to it. But of course the contributions of any one person must depend in a very essential way on the cumulative contributions of older generations of mathematicians. The work of Niels Henrik Abel provides a necessary background for a great deal of present day mathematics. The groups studied by Sophus Lie are important in many branches of mathematics. We learn not only from our mathematical ancestors, but also to a great extent from our contemporaries. I have personally benefited from the work of a number of the previous Abel Prize winners. The thesis of Jean-Pierre Serre provided a foundation for nearly all subsequent work on homotopy groups. His beautiful Cours d'Arithmétique taught me about the quadratic forms which play an important role in understanding the topology of manifolds. In fact, this study of quadratic forms was so addictive that I spent some years studying problems in algebra for their own sake. Michael Atiyah's work on K-theory provided the inspiration for my own work on algebraic K-theory; while John Tate helped me to understand the relationship between algebraic K-theory, quadratic forms and Galois cohomology. One great advantage of the long mathematical life which I have enjoyed is that it has enabled me to see amazing progress by others on problems which I had helped to formulate. Thus the work on quadratic forms which I just mentioned led to conjectures which were later verified in very deep work by Vladimir Voevodsky. Similarly, Misha Gromov's work on the growth of finitely generated groups went far beyond anything which I had been able to achieve.

**10. Fellow of the American Mathematical Society (2014).**

For contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems.

JOC/EFR July 2015

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