A standard resource for this material is T. Y. Lam's book on ``Serre's Problem on Projective Modules''. While reading this book, I realised that this is a topic on which Indian Mathematicians, mostly working within India, have made many significant contributions. (The fact that Serre's problem was actually settled by Quillen and Suslin, who are not Indian is only marginally important!) In particular, one can note that no less than 30 Mathematicians of this kind are referred to in Lam's book.

I was at the School of Mathematics, TIFR during the period when mathematicians there were in the thick of this activity. In hindsight, it is clear that I missed a golden opportunity to learn about it and appreciate it then! Still, better late than never ...

In this context, it is odd that some people think that we need to go back thousands of years to find significant contributions to Mathematics from India. All of the work mentioned above started only about 50 years ago and continues till today. Moreover, the problem (which is related to that of solving linear equations) is centuries old.

]]>(I am teaching a course on Measure and Probability and was quite happy to find the following computational exercise to give to the students.)

The following computational exercise may help clarify the different methods of integrating functions and their relative utility.

Take an any "algebraic function" *f* (like a polynomial or rational
function or something like *f*(*x*) = √(()1 − *x*^{2}) )
which is continuous on [0, 1]
.

- Integrate it using High-School methods and calculate the value. (You may need to use a power series method to integrate term-by-term.)
- Integrate it numerically by Simpson's rule (or trapezoidal rule) by choosing a sufficiently fine partition of [0, 1] into small intervals.
- Integrate it by Monte-Carlo. Choose a uniformly random large finite sequence
x_{n}in [0, 1] (by using a Pseudo-random generator) and calculating the average value off(x_{n}) .

After writing the program you should test which method is faster.

Note that if you know the function only approximately, then (1) is not available but (2) may still give a reasonable answer.

If you know only a large number of (*x*, *f*(*x*)) values
(experimental data) then (2) is also not available and you can
only use (3).

Method (1) is Newton/Liebnitz integration, (2) is Riemann integration and (3) is Lebesgue integration.

]]>Numbers play multiple roles:

- As a way of counting. (Cardinals)
- As a way of ranking. (Ordinals)

For natural (finite) numbers, these different senses of the use of numbers coincide. (Though one can have some doubts when the numbers are really large!)

In the same way, mathematical objects can be considered in different ways:

- "Pure" set-theory. (Axiomatic set theory)
- Sets with some sort of structure. (Category theory)

Even for finite sets, these notions do not coincide! For example, there are many different finite groups of the same size.

Coming to the problem of "infinity". The simplest notions of infinity are:

- The set of natural numbers. (Cardinal aleph_0)
- The ordered set of natural numbers. (Ordinal small omega)
- Asymptotic points or points at infinity. (For example, the point (1:0:0) in projective geometry)

Each of the above have associated arithmetic and algebraic operations. For example, with counting numbers we have addition and as a consequence multiplication. With ordinal numbers we have the notion of a successor which can be used to define a notion of addition. The corresponding structure in sets is that of Boolean or sigma algebra of sets. Category theory also has its own notion of algebra called ``universal algebra'', which is like (but not quite the same as) the sigma algebra of sets (infinite sums and products need to be defined and may not exist!).

So to re-phrase the question, we are asking if the ordinary notion of arithmetic and algebraic operations extends to infinity.

At first glance it does. We can certainly perform Boolean operations with infinite sets. The problem is that the usual statements about these operations are sometimes no longer true and our intuition about algebraic identities would fail us.

For example, it is usual to say that multiplication is the operation of repeated addition. When the number of additions is infinite, it is not very evident what this means. We define the product of sets AxB which clearly explains what this operation (multiplication) is for sets.

Similarly, it is natural to think of addition as repeated successor operations, but it is not always clear what this means for the infinite successor operation. Again, ordinal succession is defined in a way that such an operation is meaningful through the notion of limit ordinals.

However, in each case some "obvious" results from the finite case are no longer valid.

It is worthwhile to extend notions from the finite to the infinite when this is useful in giving us expectations regarding questions (about finite sets!) that we could not have arrived at otherwise. (For an interesting example, have a look at the Goodstein sequence.)

As an addendum, I would like to add the naive re-statement of Skolem-Lowenheim.

Since language consists of countably many sentences, we can only hope to define countably many things and from a practical point of view we can only define finitely many things.

Thus, infinity is a notion that mathematicians handle with care, limiting the roles that it can take, so that playing around with infinity gives meaningful (and correct!) results about the finitely many things that we will actually encounter! Mastering this way of handling infinity with care is what a lot of mathematical training is about.

]]>

The kind of "popular mathematics" that interests me is more in
the sense of "popular mechanics" or "popular electronics".
Something that gives the reader new things to do and helps build
tools to explore further. Just as an essay on Newton or Hook is
not an article on popular mechanics, and an essay on Marconi or
Baird is not popular electronics, so a book like that by E. T.
Bell on the lives of famous (dare I say "great") mathematicians
does not constitute "popular mathematics". (For example, this
essay is *not* (yet!) popular mathematics.)

Another aspect of "popular mathematics" is that mathematics should be its own narrative. An article on "building your own rubber-band powered airplane" that spent a lot of time discussing the joys and benefits of flying, would not interest me. In the same sense, an article that extols the beautiful applications of what one is about to study is wasting time in getting to the point --- if it is beautiful it will speak for itself.

There is certainly some interest in how the ideas (might have) come about. A pure recipe, which says, do this calculation followed by this substitution to have this formula and "Voila!" is meant for computers --- not human beings. "How does (did) one think of the steps that are to be carried out?" is relevant to the story. This need not be a factual/historical account as long as it is plausible and interesting!

As an example, consider the following (brief) account as how one can build up the notion of complex numbers from geometry.

In the study of the geometry of a (straight) line, one encounters numbers via translations to the left and right (traditionally, positive and negative respectively) which corresponds to addition of numbers. One also encounters scaling up or scaling down which corresponds to multiplication and division. What happens when one considers that plane?

One can certainly translate the plane in various directions. We can also rotate the plane and scale it. The recognition, that (after fixing a centre/origin and a unit "x" direction), each point in the plane can determine a rotation+scaling as well as a translation, allows us to define a multiplication as well as an addition of points on the plane. This operation creates the arithmetic operations making up the complex numbers.

One can also follow this up to see how regular polygons
correspond to cyclotomic numbers and how constructible numbers
involve solutions of quadratic equations. This can lead to a
*derivation* of the recipe for the construction of a
regular pentagon --- something which is rarely done in high
school geometry books.

There are number of good books on popular mathematics:

- Mathematics for the Millions, by Lancelot Hobgen
- Figures for Fun, by Yakov Perelman
- Geometry and the Imagination, Hilbert and Cohn-Vossen.
- Mathematician's Delight, by W. W. Sawyer
- Godel, Escher, Bach, by D. Hofstader

There are a number of good books which are *not*
popular mathematics:

- The Man Who Knew Infinity, by R. Kanigel
- The Men of Mathematics, by E. T. Bell
- The Code Book, by Simon Singh
- Fermat's Last Theorem, by Simon Singh

Too often (like in the current case!), one starts with the noble goal of writing popular mathematics, only to succumb to the (somewhat lazier) approach of writing about mathematical philosophy or mathematicians. This is not the only time I have done so! With faint heart, I recall an article that appeared in Science Age in 1984 on the work of Gerd Faltings --- an excellent article by Madhav Nori was vandalised by yours truly under pressure from the editors to make it "more accessible"!

]]>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.

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.

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".

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. ]]>