### Wed, 23 Mar 2011

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".

[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.