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John Willard Milnor: Abel Prize 2011

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The Abel prize for 2011 has been awarded to John Willard Milnor.[1]

In response to a query from Dr. R. Ramachandran who writes about Science and Science Policy for ``The Hindu'' newspaper group, I wrote the following short piece which others may find interesting as well.

Milnor is well-known for his work on topology and geometry. He has also made significant contributions to algebra and even number theory. In addition, he has written a number of books which are loved by graduate students in Mathematics all over the world.

One of the striking early results of Milnor was the example he gave of a seven dimensional space which is topologically a sphere but its geometric (differentiable) structure is different. This was the first example of an "exotic sphere". A nice way to state his main result (due to Ajit Sanzgiri) is that "Groups of homotopy spheres are homotopy groups of spheres".

Milnor received the Fields' Medal in 1962. In addition, the work of a number of later Fields' medallists such as Donaldson, Thurston, Mori and Voevodsky can be seen as having roots in the work of Milnor.

A more personal perspective

The first time I came across the name Milnor was when I heard that the only dimensions in which one can do algebra with division is 1, 2, 4 and 8; I was told that an "easy" proof was based on Characteristic Classes on which Milnor had written a nice book. In later years, I read a number of his other books like ``Topology from a differentiable viewpoint'', ``Morse theory'', ``Isolated singularities of complex hypersurfaces'' and ``Algebraic K-theory''. These books not only explained the results and definitions, but laid the foundations of my geometric intuition --- the same is probably true for many others in my generation.

When I joined TIFR, Raghunathan was full of praise for the work of Milnor and how his deep ideas on differential topology would "lead somewhere". One of the first lectures in our graduate seminar was by Ajit Sanzgiri on Milnor's paper on exotic spheres --- the title of the talk was ``Groups of homotopy spheres are homotopy groups of spheres''.

When Srinivas taught me (algebraic) K-theory, the only available reference text was Milnor's book (Milnor K-theory forms a crucial component of Voevodsky's early 21st century work that won him his Fields Medal!); since then Srinivas has written a more modern and comprehensive book on the topic.

Later, when A. J. Parameswaran started work on his Ph.D. under the guidance of Srinivas, AJP and I read Milnor's book on isolated singularities together --- as a prelude to the sequel to Milnor's book (by Looijenga). Much of the modern work on the algebraic theory of singularities (which forms a crucial component of Mori's Fields medal winning work on terminal three dimensional singularities) starts with the notion of "Milnor number" and "Milnor fibrations".

If the impact of a mathematician is to be measured not only by his own fantastic results but the great results of others that grow out of his work, then Milnor is certainly one of the greats of the latter half of the twentieth century.

Much of the topology and geometry that I have used in my work relies on simpler versions of Milnor's results that were proved by his predecessors. So (unfortunately!) I cannot quote a result that I have proved which actually uses a theorem of Milnor's.

``Groups of homotopy spheres are homotopy spheres''

I think the idea of joining two spaces by a tube to make a new space (called connected sum) is due to Whitney or Kervaire. This leads the "algebra of spaces" or to give it its mathematical name "the cobordism group". Milnor showed us how to perform calculations with this group.

Another group studied by topologists combines the different ways in which a sphere of one dimension ``wraps around'' a sphere of a (possibly) different dimension; this is called a homotopy group of the second sphere.

Milnor's brilliant idea was to show that in certain cases, the two groups obtained are the same. Said pithily, "Groups of homotopy spheres (exotic spheres)" are the same as "homotopy groups of spheres".

Some remarks

A number of people have remarked that what we need in order for institutes like IISER to grow are ``teaching researchers''. The system of universities in the USA has thrown up a few shining examples --- Richard Feynman in Physics and John Milnor in Mathematics are two names that immediately spring to mind.

[1]The last time I wrote about the Abel Prize was to contribute to a guest post to Rahul Basu's science blog: The Far Side. Unfortunately, Rahul is no longer among us. This post is dedicated to the memory of Rahul Basu.


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