What Is Research?

May 9, 2007

Think structures, not numbers

Recently, while I was preparing for and delivering a talk on The story of the symmetric group, I came across an interesting maxim of serious combinatorics: think in terms of sets, not in terms of numbers. For instance, the person thinking in terms of numbers says that cardinality of A is cardinality of B plus cardinality of C, while the person viewing things in terms of sets will say that A is a disjoint union of B and C. Similarly, while the number-viewer simply sees one number as a product of two others, the set-viewer will see one set as the Cartesian product of the other two.

The main advantage of dealing with sets, as opposed to dealing with numbers, is that when we have sets, we can consider maps between sets, we can look at elements of sets, and we can do all kinds of funny things which we could notdo with numbers. For instance, sayingthat the number of permutations is more than the number of unordered set partitions, is not as informative as actually describing a surjective map from the set of all permutations to the set of all set partitions. That surjective map will not only show that there are more permutations than set partitions, but will also show that the number of permutations with a given unordered integer partition as cycle type, is more than the number of permutations with that same unordered partition as subset sizes. It can thus also show that unsigned Stirling numbers of the first kind are always bigger than the corresponding Stirling numbers of the second kind, and so on.

I wouldn’t have paid too much attention to this paradigm of thought that I observed myself had it not been that, while discussing Olympiad mathematics with a young aspirant, Ashwath, I happened to repeat these ideas and he really took them to heart. He said that this aprpoach really changed the way he looked at combinatorics, and helped him perceive much more structure in it.

Over the last few days, I have seen a lot of situations wherein I am forced to revise my Think in terms of sets, not numbers paradigm. Rather, I should say think in terms of structures, not numbers where structure usually means a set with additional combinatorial, algebraic, topological or other information encoded into it. Those who are thinking as primitively as numbers, of course need to move to the set-level. But even those who think only in terms of sets need to move upwards.

In fact, it is illustrative of all that I am saying that many laypersons describe mathematics as the study of numbers, while most mathematicians define mathematics as the study of structures and patterns.

I’ll describea few examples.

One of the recent developments in mathematics is the so-called Khovanov invariant theory. Prior to Khovanov invariants, the best way to study a knot (yes, knot in the usual physical sense) was to associate to it a polynomial, the so-called Jones polynomial, whose coefficients basically store some numerical information obtained by trying to perform Reidemeister moves at ea ch crossing of the knot. The Jones polynomial is thus essentially a set of numbers associated with the knot.

Khovanov came up with the brilliant idea of modifying the original construction so that, instead of numbers, we manage to get whole families of vector spaces, such that if we take the dimensions of these vector spaces, and perform suitable summations, we land up with the coefficients of the Jones polynomial. The beauty and power of Khovanov invariants lies in the fact that one can look at these vector spaces (they give a homology theory) and define maps between vector spaces — in fact (for those who care) this is a functor from the category of knots with morphisms as cobordisms, to the category of homology complexes with the usual morphisms of complexes.

Similarly, recently, one of the research students at the ENS, Olivier Dudas, was telling me about his research work on the representation theory of finite groups. It turns out that in order to study certain kinds of representation theory, he again needs to consider other, more complicated, complex-like objects such that, when we take suitable alternating sumations, we get down to the original object. In other words, the representations themselves do not have enough structure — it is necessary to go beyond them to some more structured object to draw inferences and conclusions.

The great (as well as irritating) thing about mathematical structures is that they live on for ever. Once we create a mathematical structure, there is no way of destroying it, except through creating an even more inclusive mathematical structure. Thus, mathematical structures that were created or devised to solve particular problems, have stayed on in history and have soon been studied in their own right.

The recent development of the theory of quantum groups illustrates yet another principle in this structure-oriented thinking. Namely, that if one wants to consider variation of a structure, which itself is fairly rigid, it makes sense to put a lot of other auxilliary structure, and then vary. For instance, onecannot directly vary the structure of a group — it is a fairly rigid object. HOwever, what one can do is look at the group algebra and try to geenralize the notion of group algebra to that of Hopf algebra, and then try to deform the group algebra slightly in such a way that it continues to remain a Hopf algebra. This gives us the notion of quantum group.

It is rather unfortunate that in our school education, even the elementary language of sets is taught only in high school, and is not used to its fullest power even in the most elementary of combinatorics. The strange thing is that set-theoretic language is more concrete than the numerical language, so it should be what the champions of concrete mathematics should favour — ideally every combinatorics problem should be solved with sets and cardinalities should be taken at the very last step, for computations. If students are inculcated with this mindset, it may be much easier for them to grasp the structural thinking of higher mathematics.


May 8, 2007

A summer in Paris — the ENS

Filed under: ENS,Places and events,Regular updates — vipulnaik @ 1:48 pm

As part of an exchange programme between Chennai Mathematical Institute and the Department of Mathematics at Ecole Normale Superieure, three of the people in my batch (Shreevatsa, Arul and I) are spending two months at the ENS. We are living at the Montrouge quarters of the ENS, and our academic headquarters (so to speak) are at the main ENS in Rue d’Ulm.

Neither CMI nor the ENS has placed any academic expectations on us. They have basically given us some facilities and have asked us to fend for ourselves, making use of these facilities. The general plan is that each of us study some topic(s) under the guidance of ENS faculty, and we may possibly be asked to present what we have learned at the end.

Prior to the ENS, I have studied/interacted with faculty for a long period of time, at my own college (Chennai Mathematical Institute), the Institute of Mathematical Sciences, and the Tata Institute of Fundamental Research. Each of these places was very different in terms of size and atmosphere. CMI is a rather small and informal place — it has almost nobody except students and faculty (that is, very little administrative staff), and its only departments are mathematics, computer science, and physics. All the offices have glass doors and open out to the grounds. The Institute of Mathematical Sciences is a relatively larger place, with an often irritating central air-conditioning, many more office rooms, and a much more closed look to it. Though the departments are the same, the sizes are much more. There are big roms with coffee-table discussions. There are a whole lot more administrators, and the place in general boasts of a much bigger size than CMI.

Tata Institute of Fundamental Research is truly monumental compared to CMI, with departments including Mathematics, Physics, Computer Science, Theoretical Physics, Chemistry etc. Apart from the large number of academic faculty, there are a whole lot of administrators. There are huge living quarters in addition to the main institute building. Overlooking the sea, TIFR is both open and closed — open in the sense that the rooms open out to the sea, closed in the sense that it’s a centrally air-conditioned building and one can shut the outside world and concentrate.

That said, all these places had some overall similarities: the way in which students and faculty members interacted, the kind of food, the way people organized themselves, was quite similar. The Ecole Normale Superieure is proving to be somewhat different.

Unlike CMI, IMSc, or TIFR, the ENS is located pretty close to the center of the city; not that this says much, because the center of Paris is not as crowded or congested as the center of an Indian city. However, it probably reflects the general trend in Paris to have universities everywhere, not just in far-away isolated corners. The ENS has departments in sciences as well as humanities and has a total of over a thousand students, including both students completing the last three years of their five-year diploma (the French equivalent of a B.Sc. cum M.Sc.) and research students.

The mathematics department itself has some 60-70 faculty members as well as many other visiting faculty from institutes like Orsay.

One of the striking features of mathematics at the ENS (at least to a person who’s studied in India) is that most of the mathematics here is done in French. In fact, almost all discussions amidst students and faculty members is in French, and courses and talks are mostly in French. Talks are in English only when the speakers come from other countries (which again may not necessarily be English-speaking). This often leads to some interesting language problems and issues. For instance, to publish in journals outside France, one must write in English, and to learn about cutting-edge work done outside France, one must read English. Thus, most of the older graduate students, as well as faculty members, speak fairly good English, and can lecture in and understand English.

I was not completely taken unawares by this because during the International Mathematical Olympiads in 2003 and 2004, I had seen people from different countries write the Olympiads in their own respective languages — the Hungarians wrote in Hungarian, the Chinese wrote in Chinese, the Japanese in Japanese and so on. The only countries which wrote the IMO in English were India, Sri Lanka, Trinidad and Tobago, US, UK, and some Arab and African countries — basically, countries which imported much of modern mathematics from outside.

Aside from the language, another thing that greatly impressed me about mathematics at the ENS (fr whatever little I have seen about it) was the great professionalism and care with which people talked while lecturing. This may in part be due to the system of French education, where great emphasis is placed on presentation skills and where students are grilled orally by instructors on a regular basis. I hoep to understand better how the French present stuff by attending some talks here at the ENS — if they are in French, that’ll also be an opportunity for me to try deciphering French in real time.

Another nice thing about the ENS is its library (or bibliotheque, as it is called in French — the word librarie is used for bookshop). The library is pretty huge, with a lot of books both in English and in French. It also has an interesting system of organization (which I have not yet cracked) and a lot of helpful librarians). The place is also maintained in a way that a lot of people can do a whole lot of serious study there — and the librarians are very helpful with locating stuff.

Now as to my academic programme.

Dr. Olivier Glass, the academic coordinator for the exchange programme, told Arul and me (the two who are interested in mathematics) to contact the faculty members David Madore and Olivier Schiffmann.

Dr. Schiffmann sent us a list of possible topics which we could study over the summer, which included Schubert calculus, removing singularities, quantum groups, representations of quantum groups,quivers and Hall algebras, and Khovanov invariants. All the topics were very interesting, so Arul and I met Dr. Schiffmann on Monday (7th) and he told us a little bit about each topic. I enjoyed all of them and for some time was in a dilemma as to which one to choose. After some thought, i decided to pick on Schubert calculus, because I had been studying stuff on related lines for some time and I thought this would be a natural extension of that stuff.

I was and am also keen on studying quantum groups and I shall probably be going over to these if I am able to reach a point of closure with Schubert varieties.

Will keep posting as I get more and more of an idea of the life at ENS.

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