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The closest thing number theory has to a Grand Unified Theory is the Langlands program. It's a huge, highly interconnected, often surprising bunch of conjectures relating number theory to geometry, algebra, and even[s] string theory -- and also something that I know next to nothing about. :)

I'll put the Wikipedia[w] intro here, and note with some happiness that Frenkel has used that exact phrase before:

> In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest projects in modern mathematics, Edward Frenkel described the Langlands program as “a kind of grand unified theory of mathematics.

Wiles' proof of the modularity theorem (and hence Fermat's Last Theorem) is a proof of a some of the Langlands conjectures "in dimension 2": one of the pinnacles of human mathematical achievement is just an incomplete portion of a special case of Langlands' vision.

> Andrew Wiles' proof of modularity of semi-stable elliptic curves over rationals can be viewed as an instance of the Langlands reciprocity conjecture, since the main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture for GL(2, Q) remains unproved.

To give you a little taste, here's a 10000-miles-above-the-ground view of what this says.

A modular function is a certain kind of object in complex analysis (something like calculus with complex instead of real numbers[c]).

A Galois representation starts with a Galois group, which tells us something about the large-scale structure of a number system, and extracts[p] ("represents") a small part of the information it contains: the full "absolute" Galois group is incomprehensibly complicated. Every elliptic curve gives rise to these representations.

There's no reason these things should be related: most number thought the modularity conjecture was an improbable goal. Now think about how it's just a tiny part of a much larger whole. They don't call it "fantastically bold"[n] for nothing.


[w]: https://en.wikipedia.org/wiki/Langlands_program

[s]: https://physics.stackexchange.com/questions/4748/why-is-ther...

[n]: https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan...

[p]: As I understand it, Wiles' proof only uses 2-dimensional representations.

[c]: ... but it's hard to overstate how much nicer and more uniform everything becomes!

I would also recommend the book by Edward Frenkel, "Love and Math" where he talks about his involvement in the Langland program.

Also, he has a very cool lecture on the subject here:

"What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?" http://online.itp.ucsb.edu/online/bblunch/frenkel/

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