Had a quick look at elliptic curves: Fascinating subject, but it went quickly over my head -> I have the standard physicist training in analysis and (linear) algebra and am familiar with elliptic integrals, complex analysis and elliptic functions (to a degree), though my 'mathematical horizon' is pretty much Lie-groups/algebras (and my bag of 'what I picked up along the way' usual for physicist). Pretty much the *"It's neither differentiable, nor combinatorics, so why bother?"*-attitude to 'discrete stuff'. Guess that was a bit ... premature.

Lie groups/algebras! Were you doing stuff on elementary particles, supersymmetry and the like? My own dissertation was on algebraic groups (= positive characteristic analogues of Lie groups) where the Lie-algebra side is the same (but won't give quite as conclusive results as in the boring characteristic zero case). We use algebraic geometry there as a subsititute for analysis. Thankfully you only need to believe in (the results of) algebraic geometry. Something I could do even though I never quite got the hang of AG (other than in the simple case of curves). Equally thankfully familiarity with curves allows you to have fun in EC crypto as well as in error-correcting-code side - the latter I have worked on more seriously.

No, I work in theoretical solid state physics; my toolbox is

*statistical quantum field theory* - but that means I'm 'only a Wick-rotation away' from many of the ideas and concepts from high-energy/particle physics & there's always been an overlap between the two fields (cf. Feynman, Dyson etc.). We also get the basic training in

"second quantization" that the high-energy/particle physicists get, and I'm 'comfortable' with most of quantum electrodynamics, but I have no serious training e.g. QCD.

I have no

*formal* training in Algebra - but Lie Algebras are so ubiquitous in quantum theory ((bosonic) Commutator, Poisson Bracket, outer vector (cross-) product, SO(n), SU(n), Campbell-Baker-Haussdorff relation ...) that you 'know' how the gears work long before you pick up a book to

*"let a mathematician confuse you about stuff you already use on a daily basis"*.

(Especially as a theoretical physicist, you have to take care not to

*'drown in the mathematics'*, because there's

*"Just so much Shiny!"* - technically, my toolbox touches on pretty much three fourths of the hot topics of mathematics of the last two centuries - you need to get a working knowledge on what you need to do your job rather than an exhaustive insight into every subject.)

It helps a bit that German physics undergraduates going the 'Diploma'-curriculum path (the system before we adopted 'Bologna') trained together with the mathematicians for about 4 Semesters - we pick up a

*'algebraic structures emergency field-kit'* already in the first two Semesters, so to speak, but our knowledge is application-oriented, rather than rigorously formal. Wrt. Algebra, you're often introduced to the general ideas using specific matrix-representations of generators (e.g. Pauli-matrices) and over time, you see that certain stuff just keeps showing up again and again ...

My speciality is strongly correlated 1-dimensional systems & I'm dabbling in topological isolators/crystals -

Luttinger Liquid theory and bosonization are my weapons of choice.

Andew Wiles did study elliptic curves, but the machinery he used goes way over my head. I once had a colleague who wanted to work on Wiles' proof. He got to something like the half-way point (that's what he said), but then had to quit. The poor guy never finished his PhD. He took up teaching and running, and went on to win the 50+ class at Berlin Marathon!

Didn't know that Shor would have a quantum computer algorithm for discrete logs? God, I'm out of touch of what's going on. Years of teaching calculus to reluctant physics majors and bottles of fine single malts are taking their toll

Hey man - you

*do* understand I'm riffing off of a Bio of Wiles' I once read & the three jargon-buzzwords I picked up from

~~sleeping through~~ sitting in the talks of the quantum-CS department, do you? (Jeez, you Mathematicians are so easy discombobulate ...

). The stuff about Shor's discrete log algo I picked up on Wiki, btw ...

IIRC, "half of Wiles' proof" is already the equivalent of two or three Masters-level courses plus a dissertation or three? I heard that the beauty of it was that Wiles connected two (three) different fields of research so rigorously that Fermat's conjecture simple falls into your lap as a byproduct in the end - but that it's applications go far, far beyond it?