The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection of tricks used by all mathematicians. That question now has many answers fitting this "there exist a small set of tricks used by all mathematicians" interpretation. I find that swapping the quantifiers gives a better question. I.e. I am more interested in hearing about the small collections of tricks of individual mathematicians. Pointing back to the other question above, and Rota's article, what are the few tricks of Erdős, or of Hilbert?
Question: What are the few tricks of some individual mathematicians?
Of course, as the comment in the earlier question quips, a mathematician never reveals tricks…but one can hope. In your answers, please include the name of the mathematician, and their few tricks…perhaps some cool places where the tricks are used, i.e. some "greatest hits" applications of the tricks.
Note, I don't think that knowing these tricks can make you into Erdős or Hilbert, but a long time ago a friend told me that a talented mathematician he knew would approach research problems by asking himself how other mathematicians would attack the problem. This is sort of like writing in another author's style, which can be a useful exercise. Wouldn't it be neat to be able to ask yourself "How would Hilbert have attacked this problem?"
MO is a good place to collect these, because it often takes extended reading (as intimated by Rota) to realize the few tricks used by a certain mathematician. As a community, we may be able to do this.
Best Answer
The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it's OK if I specify my own "one weird trick". The entirety of my research centres around the idea that, if $\chi$ is a non-trivial character of a compact group $K$ (understood either in the sense of "homomorphism to $\mathbb C^\times$", or the more general sense of $k \mapsto \operatorname{tr} \pi(k)$ for a non-trivial, irreducible representation $\pi$ of $K$), then $\int_K \chi(k)\mathrm dk$ equals $0$.
It's amazing the mileage you can get out of this; it usually arises for me when combining Frobenius formula with the first-order approximation in Campbell–Baker–Hausdorff. Combining it with the second-order approximation in CBH gives exponential sums, which in my field we call Gauss sums although that seems to intersect only loosely with how number theorists think of the matter. Curiously, I have never found an application for the third-order approximation.