Algebraic geometry is the purest of pure branches of mathematics, concerned with the intrinsic structure of functions. Its central tool is polynomials, which constitute the very special case of functions you can “write down”.
Chebfun is the most practical of practical branches of mathematics, concerned with machine computing with functions. And its central tool is polynomials too! Mine is the kind of mind that needs to know, are algebraic geometers and Chebfunners interested in polynomials ultimately for the same reason, or different?
I think they are different. The starting point of algebraic geometry is that, given a function f and a point a, there is a polynomial that exactly matches f and its derivatives at a. The starting point of Chebfun is that, given a function f and an interval [a,b], there is a polynomial that approximately matches f on [a,b] to any prescribed accuracy.
[9 October 2014]