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Some other results in Geometry that do not require reaching very far to see its connection to PDEs: the resolution of the Yamabe Problem, the proof of the Calabi Conjecture (now the Calabi-Yau theorem), and the proof of Positive Energy Theorem.

(I violate the 1 example per answer rule, since these three are all from geometry, and involve the same mathematician.)

Edit: As Deane pointed out below, I should be more precise about the attribution. A well known contributor to the solution of those three problems above is S.T. Yau. Others who have worked on those problems include Rick Schoen, who collaborated with Yau on the proof of the Positive Energy theorem and (hence) the Yamabe problem, and Thierry Aubin who also contributed much to the understanding of the Yamabe Problem, as well as making significant progress toward the Calabi conjecture.

Edit 2: And of course, as Timur pointed out below, I inadvertantly left out Neil Trudinger as one of the main contributors to the Yamabe problem. (One of the reasons I didn't want to be too precise on references in the beginning was to avoid mistakes like this.) Also please note that this is a Community Wiki article, so please feel free to just edit it to fix any insufficiencies you see.

2 added 487 characters in body

Some other results in Geometry that do not require reaching very far to see its connection to PDEs: the resolution of the Yamabe Problem, the proof of the Calabi Conjecture (now the Calabi-Yau theorem), and the proof of Positive Energy Theorem.

(I violate the 1 example per answer rule, since these three are all from geometry, and involve the same mathematician.)

Edit: As Deane pointed out below, I should be more precise about the attribution. A well known contributor to the solution of those three problems above is S.T. Yau. Others who have worked on those problems include Rick Schoen, who collaborated with Yau on the proof of the Positive Energy theorem and (hence) the Yamabe problem, and Thierry Aubin who also contributed much to the understanding of the Yamabe Problem, as well as making significant progress toward the Calabi conjecture.