I think Deligne's Theorem is a poster child for the power of Modern Algebraic Geometry. Andrew Wiles's proof of Fermat's Last Theorem might also be relevant. G

I think Deligne's Theorem is a poster child for the power of Modern Algebraic Geometry. Andrew Wiles's proof of Fermat's Last Theorem might also be relevant. Generally speaking, studying what the solution sets of polynomial equations is "like" is quite fundamental to a lot of mathematics. Doing this in a "deep" way can lead to a reimagining of much of modern mathematics: https://rawgit.com/iblech/internal-methods/master/notes.pdf https://rawgit.com/iblech/internal-methods/master/notes.pdf [edit] For instance, lots of people use a straightforward generalisation of number systems called rings. But ring theory is quite abstract. Modern Algebraic Geometry shows that at least in the case of commutative rings, these are merely spaces of functions on a space called a ring's spectrum. You can visualise a ring's spectrum, unlike the ring itself. Many properties of a ring are just properties of its spectrum. This seems like a significant conceptual leap in the understanding of things that were studied since the 1800s without much geometric understanding. Oh yeah, and I'm not an algebraic geometer.