The two distinct cultures in mathematics

Question

I once read an article about two distinct culture within mathematics, "analysts" and "algebraists" when I was in high school. I am not still a graduate student, but I really want to hear what it feels to be an analyst or an algebraist. How different their tools are in research? And how different their takes are on the subject? Please share your experiences :)

Answer 1

Here's a very soft answer to a very soft question: Are you familiar with the psychological notion of Rorschach inkblot images, designed to probe the human subconscious ?

When exposed, for instance, to the exact same pictures of various polynomial shapes, the algebraist mind immediately asks itself: Hmmm... I wonder if the curve has any rational points..., whereas the incorrigible analyst instantly thinks to himself: Hmmm... I wonder whether the arc length or area don't by any chance possess a closed form expression... Time to pull those integrals out of the trunk... :-) For the algebraist, $x^n+y^n=z^n$ means Fermat's last theorem and Beal's conjecture; for the analyst, it's a superellipse. The same object is approached from two different perspectives. Where the algebraist sees Wiles, the analyst catches a glimpse of Lam$\acute e$.

Answer 2

Well I don't think there is a clear borderline. Usually to a classical algebraist structure is more abstract in the sense that you throw in one or (more )binary operation/s in a set and go from there. You study the substructures, try to classify all the substructures with certain properties etc. Topology does not matter much. But to an analist without topology (e.g norm, weak etc)space is not rich enough for certain applications. Proximity and tools to measure "distance" between different objects play a big rule.That all being said modern theories basically combine ideas of the two.for instance you can read about Analysis on locally compact groups, or lie algebras or even algebraic geometry. Its sometimes so difficult to distinguish that thin borderline. For slightly Unorthodox approaches to analysis I can name S.Banach, Gelfand and Grothendieck who passed away a few days ago.