A lot of enphasis in pure mathematics in placed on beautuful and instructional proofs, and for good reason. However, in some circles the quest for elegant formalism can come at the cost of neglecting the art of computation.
As a graduate student this feels like an actual issue: the culture in many pure mathematics fields seems to almost look down on work devoted to computations and examples for their own sake. I think this is a devastating flaw of modern pure mathematics.
Don’t get me wrong, I know there are many great authors who devote their lives to some of the most gruesome and admirable computations imaginable. My feeling is that by and large my generation of mathematicians feel compelled for one reason or another to avoid a direct computational style and instead focus on all encompassing abstraction.
There are plenty of questions out there asking for the most beautiful formulas and proofs in mathematics.
I ask: in your opinion, what are some of the most elegant computations in the pure mathematics literature? What are some examples in papers which you feel have completely changed how you view a particular field?
As an example of a computation which I have in mind, I would consider Kontsevich’s enumeration of rational plane curves a beautiful computation. I would also nominate the computation of the algebraic K theory of finite fields as another example.
This question seems similar to an old MathOverflow question about fundamental examples. Some of the answers to that question involved some significant amount of computation (e.g., the ones involving fractals or dynamical systems), and could serve as answers to the present question.
As a clarifying point, I'm assuming that the question is not about elegant algorithms. That question is a popular topic in some quarters, and if you Google “top 10 algorithms” then you'll get some candidates. It sounds like you're interested in specific instances of computation—a single run of an algorithm, one might say—where both the computation itself and the result were striking.
In the same spirit as your Kontsevich example, I would say that the original mirror symmetry computations by physicists such as Brian Green were beautiful. Indeed, physics is an excellent source of beautiful computations. For just one example, how about Onsager's computation of the Ising model?