The idle ramblings of a Jack of some trades, Master of none

May 27, 2007

The Mathematical Gene?

The high-point of my mathematical creativity occurred in the third year of college. Our lecturer of mathematical analysis, Ms Nandita Narain, had posed the following question: if a differentiable function f : R -> R is such that f(a + b) = f(a) * f(b) for all a, b in the reals, then show that f has to have the form f(x) = exp(cx) for some constant c.

I managed to prove it for continuous functions f, an even more general class than merely differentiable ones. Much chuffed as I was, that was the last spark of my mathematical gene.

Unlike me, however, such notables as my buddy Eknath Ghate have gone on to greater and better things in the world of mathematics.

Consider, for instance, the following abstract of a recent talk that Eki presented at the University of California at San Diego: Ordinary forms of weight at least 2 give rise to locally reducible Galois representations. Greenberg has asked whether these representations are semi-simple. One expects this to be the case exactly when the underlying form has CM. We shall speak about various results towards this expectation that use p-adic families of forms and deformation theory.

What a world of difference from undergraduate mathematics! To reach this stage of proficiency, it took Eki twelve years of school, four years of baccalaureate studies, five years of doctoral work, and another seven-odd years of intensive investigation in the relevant fields. Presented with this abstract, even a bright undergraduate would be stymied. Is it any wonder that the average man on the street is utterly befuddled by mathematics?

Popularisers of mathematics, such as Keith Devlin, recognise that it is impossible even to state its greatest unsolved problems in a fashion comprehensible to a man on the street. Nobel-winning work in medicine or physics can be described superbly and compactly in a few pages of the Scientific American. But mathematics, that most abstract of sciences, has now reached such a stratospheric level of esoterica, that it is pretty much useless to try an explain it to a non-specialist.


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