I suspect one reason algebraic geometry comes across as sexy is that so many fields medals went there in recent decades. But in my own experience, it seems to draw people in by its "sweep". When I went to UGA in 1977 I was almost the only person in algebraic geometry. Some years later there were so many people claiming to be in algebraic geometry in some form, people who previously announced themselves as specialists in some other area, that as a joke, my friend, an operator theorist, announced himself on the day we introduced ourselves to each other, as an algebraic geometer.
In practical terms, it turned out that algebraic geometers at UGA were so broadly trained and interested, that they could work together with many other people. Collaborative efforts developed between algebraic geometers and algebraists, number theorists, differential geometers, complex analysts, geometric analysts, even applied mathematicians. Algebraic geometers, at least the ones I know, also know at least something about algebra, algebraic topology, representation theory (not me), differential geometry, several complex variables and complex manifold theory, number theory, and some also are experts in logic. People with questions often showed up in the offices of algebraic geometers to have them answered, even if sometimes they closed the doors first to conceal that they were asking.
I once had a colleague in functional analysis come to me excited about his recent results on something he apologetically said was some abstruse concept called "Fredholm operators". I was puzzled that he thought I would not know what this was, as it was very familiar to me, namely an operator on a Banach space with finite dimensional kernel and cokernel. As is well known, certainly to "all" algebraic geometers, these are basic examples of operators with a well defined concept of "index", namely the difference of the dimensions of those two subspaces, as well as the fact that this index is constant on connected components of the space of such operators. Basic theorems in algebraic geometry and global analysis, (Atiyah-Singer Index theorem), concern giving topological formulas for such indices, and the Riemann - Roch theorem is a classical precursor of these results. Indeed the famous topic of K-theory, developed by Grothendieck in connection with his generalized Riemann - Roch theorem, involves both Fredholm operators and the space B* of units of the Banach algebra B of all bounded operators on a separable complex Hilbert space; i.e. both those spaces, Fred and B*, are "classifying spaces" for K theory, (see K-theory, appendix, by Michael Atiyah). So algebraic geometers tend to know something about Banach algebras and Fredholm operators even if (some) functional analysts do not know what K theory is.
You probably know that Grothendieck, the most impactful algebraic geometer in a century, started out in functional analysis. Of course as noted just above, it is well known that in both subjects, one recovers a space from the algebra of functions on that space by taking the space of maximal ideals in that algebra, or more generally in scheme theory, prime ideals.
elementary exercises: there is a one-one correspondence between the maximal ideals of the ring of continuous functions on the closed interval [0,1] and the points of that interval.
there is a one-one correspondence between the maximal ideals of the polynomial ring C[X], and the space C, where C is the complex numbers.
less elementary: these correspondences hold also for continuous functions on compact hausdorff spaces, and polynomials on affine spaces C^n of any finite dimension. In both cases they are given by sending a point of the space to the maximal ideal of functions vanishing on that point.
In fact compactifications of a locally compact Hausfdorff space X correspond to constant - containing, point - separating, uniformly - closed, subalgebras of C(X).
(I hope I have this right, it has been over 50 years since I did the functional analysis exercises. I remember thinking it was fun to imagine which sub algebra compactifies an open disc as a closed disc, or as a sphere, or as projective 2-space.)
Remark: As to the influence of functional analysis on abstract algebraic geometry, Hilbert proved the algebraic geometry result above (Hilbert's nullstellensatz) in 1893, (Mathematische Annalen, 42 Band, 1 Heft, p.320), 20 years before the birth of Gelfand, who is often associated with its functional analysis counterpart.
In my case, before coming to algebraic geometry, I studied functional analysis, differential topology, algebraic topology, commutative algebra and (derived) functors, and several complex variables; none of it was wasted in the end. I wound up working in complex algebraic geometry, and am now trying to learn scheme theory, in retirement.
So , maybe today algebraic geometry is just seen as a very big tent, and lots of people shelter under it.
Speaking of a big tent, I was thinking one topic I knew nothing about was physics, and then remembered it depends what you consider as physics. I was once invited to deliver a series of lectures on Riemann surfaces to a conference of string theorists, who had decided that a Riemann surface should be considered an elementary particle! I also think of pde as foreign territory, but recall that the key result of Hodge theory, which I have studied (in the context of presenting Kodaira's proof of his "vanishing" theorem), is the representability of deRham cohomology classes on complex manifolds by "harmonic" forms, i.e. ones satisfying the Laplace equation. And the key ingredient of the theory of Jacobian varieties of complex curves is Riemann's theta function, a fundamental solution of the (several variable) heat equation. So the only basic one I have not consciously run across is the "wave equation".
By the way, if you think you don't like algebraic geometry, you might take a look at Semple and Roth, or Milkes Reid's Undergraduate algebraic geometry. I myself find my eyes glaze over when I peruse derived functor cohomology of sheaves, but am fascinated by exploring the structure of the 27 lilnes which lie on any smooth cubic surface in complex projective 3 space. I am even more magnetized by constructions like trying to see how those lines specialize when the cubic surface degenerates to three planes. I.e. If S is a smooth cubic surface and F is the union of three planes, consider the limit of the lines on S in the family F+tS as t-->0. Note that S meets each of the 3 lines where pairs of the planes of F meet, in 3 points. See if you can see why, as S approaches F, a line L of S must come to lie in one of the 3 planes of F, say ∏, and since the other two planes of F meet ∏ in two lines, M and N, the limit of L must join one of the three marked points of M to one of the three marked points of N. This gives all 27 limiting lines, 9 in each plane. For help, consult the book on lines on the cubic surface, by Beniamino Segre. I.e. to care about the modern formalization of algebraic geometry, it helps (me) to know some of the beautiful results that one wants to make precise and rigorous.
Here's another example: for a complete intersection curve C in P^3 of smooth surfaces S and T of degrees d and e, the canonical sheaf on C is O(de(d+e-4)), so 2g-2 = de(d+e-4), where g = genus(C). Hence if d = e = 2, we get 2g-2 = 0, hence C is a genus one curve, i.e. a torus. This result is found in Hartshorne, p. 352, i.e. after hundreds of pages of dense theory.
Now consider this 19th century quick calculation: degenerate one quadric surface to 2 planes, which thus meet the other quadric surface in 2 plane conics, both of genus zero (we assume this for the moment), and the two conics meet each other in two points (where the common line of the two planes meets the quadric surface). Since each conic is topologically a sphere, the union of two spheres with two common points is obviously the result of degenerating a torus by pinching two circles. So before degeneration we had a torus, i.e. a curve of genus one. To see why a plane conic has genus zero, project it from a point of the conic bijectively onto the (projective) x axis.
It is of course important to know why these calculations are rigorously correct, but it is also bad form to deprive students of powerful computational tools that were known and available well before the advent of rigorous methods.
