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Archive for the ‘Combinatorics’ Category

The writings (and even papers/technical books) of Gian-Carlo Rota are perhaps amongst the most insightful that I have encountered in the past 3-4 years (maybe even more). Rota wrote to provoke, never resisting to finish a piece of writing with a rhetorical flourish even at the cost of injecting seeming inconsistency in his stance. I guess this is what you get when you have a first rate mathematician and philosopher endowed with an elegant; at times even devastating turn of phrase, with a huge axe to grind*.

The wisdom of G. C. Rota is best distilled in his book of essays, reviews and other thoughts: Indiscrete Thoughts and to some extent Discrete Thoughts. Perhaps I should review Indiscrete Thoughts in the next post, just to revisit some of those writings and my notes from them myself.
Rota in 1962

Rota in 1962

This post however is not about his writing in general as the title indicates. I recently discovered this excellent dialogue between Rota and David Sharp (1985). I found this on a lead from this László Lovász interview. Here he mentions that Rota’s combinatorics papers were an inspiration for him in his work to find more structure in combinatorics. From the David Sharp interview, here are two relevant excerpts (here too the above mentioned flourish is evident):
“Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.
[…]
Much combinatorics of our day came out of an extraordinary coincidence. Disparate problems in combinatorics, ranging from problems in statistical mechanics to the problem of coloring a map, seem to bear no common features. However, they do have at least one common feature: their solution can be reduced to the problem of finding the roots of some polynomial or analytic function. The minimum number of colors required to properly color a map is given by the roots of a polynomial, called the chromatic polynomial; its value at N tells you in how many ways you can color the map with N colors. Similarly, the singularities of some complicated analytic function tell you the temperature at which a phase transition occurs in matter. The great insight, which is a long way from being understood, was to realize that the roots of the polynomials and analytic functions arising in a lot of combinatorial problems are the Betti numbers of certain surfaces related to the problem. Roughly speaking, the Betti numbers of a surface describe the number of different ways you can go around it. We are now trying to understand how this extraordinary coincidence comes about. If we do, we will have found a notable unification in mathematics.”
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*While having an axe to grind is a fairly common phrase. I got the idea of using it from an amazon review of Indiscrete thoughts (check link above). Because I really do think that that is the best description for a lot of his writings!

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