How much of this math is deep work and how much of it is it the conjuring of obscure objects that haven't had much study and proving trivial things about them?
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
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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.
Modern Algebraic Geometry is indeed highly abstract, but generally the conjuring of obscure objects is with a specific goal in mind, for example
- consolidation of many types of results into a `simple' theoretical framework, I suppose this originates with Noether, and reaches it apotheosis in Bourbaki's tracts.
- embedding of 'classical' objects (solutions to polynomial equations) inside a larger 'category' (schemes) where certain mysterious relations observed in the classical world (Weil Conjectures) have a more `natural' interpretation (fixed point theorem) and light the way to a proof which would have otherwise been beyond reach
Algebraic geometry has many deep and powerful ideas. It started out by looking at the space described by the zeros of a polynomial, eg
y^2 - x^3 - x = 0, x^2+y^2+1=0
But actually you do not need to talk about the underlying space directly. If you want to talk about a space, all you actually need to think about are the possible functions on the space. If you want to talk about geometry, you only need the algebra of functions on that space, so in the example just the polynomials themselves, rather than having to say explicitly solve it for the points. You can use this big idea in a lot of other areas of mathematics and physics.
The whole approach of Grothendieck (the “rising sea” analogy) is to conquer deep theorems by first conjuring obscure (but simple) objects and prove simple properties of them, again and again, until finally the deep theorem is a result of many simple ones. In this project (and Grothendieck’s program) the deep result is the Weil conjectures.
I once tried to learn some math from first principles in order to solve a symbolic dice problem. I had a question deleted from mathoverflow (probably because I did a lousy job wording the open ended question of what the heck the mathematical terminology even was for the problem I was solving) but it lead down the rabbit hole of abstract algebra and into universal algebra. I had discovered a simple dice problem where the only mathematical representation was a a “commutative magma” and I found reading the surrounding material fascinating. The ideas behind universal algebra, the notion boiling mathematics down to a set of arbitrary objects and abstract and arbitrary operation(s) and classification and rules build up based on what you define/allow the rules to be.
It’s surprisingly simple at a conceptual level but I rapidly stopped researching as I found the entire field seems to assume a phd level of math knowledge and terminology. For what I probably could have grasped in high school.
Math needs more people to try and build a chain of understanding “up from the ground level” instead of arbitrary starting points based on assumptions regarding prior learning and educational pipelines/universities.
I think you may run in to an issue with the fundamental nature of math in answering that question because the path to one deep theorem is usually made up of a many obscure objects linked through trivial steps.
