Going back further, algebraic geometry over the complex numbers was shown in the early 20th century to be in many ways equivalent to more classical analytic geometry: https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...
A mathoverflow thread with cool uses of schemes: https://mathoverflow.net/questions/59071/what-elementary-pro...
But I tend to agree with his first statements about polynomials: They look too restrictive to be highly promising for physics.
It appears that he drifted off a central focus on physics and got interested in some math that may, long shot, have something to do with some detailed aspects of string theory. It looks like he is more interested in the math than the physics.
Okay, but to me it's a loss for physics.
Going back to the 1800's, in the theory of Riemann surfaces (one-dimensional complex manifolds), the only meromorphic (complex differentiable) functions are ratios of polynomials! And all closed projective complex manifolds are algebraic varieties. See https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...
This is rather a tautology since polynomials are exactly defined this way. I can imagine that if we built a (fictional) number system rather on properties of the Fourier transformation, sines and cosines would be very natural operations that would have a deep generelization when sufficiently abstracted.
On the other hand, by restricting ourselves only to polynomials and their solution sets, it turns out that the singularities which arise are not too bad, and we can study them in detail. In other words, restricting to polynomials is the _only_ restriction we have to make, pretty much every solution set can be studied from there on.
Like how we kept ignoring nonlinear differential equations, because they aren’t that well-behaving as their linear counterparts... And then when we eventually looked into it, we found a completely new paradigm: chaos theory.
All the different areas of math just start at different places to reach toward the same important questions. It's like martial arts: aikido, shotokan, capoeira, krav maga -- very different approaches and understandings, but the masters are all reaching for the same goals, coming from different foundations and directions.
Fascinating article about the geometry of complex numbers: https://acko.net/blog/how-to-fold-a-julia-fractal/
I don't want to discourage you, I'm just being realistic. If you want to work towards algebraic geometry, you can certainly do that. You'll need to first master linear algebra and abstract algebra. You should have a strong understanding of fields, groups, rings, vector spaces and modules. Someone else mentioned commutative algebra - that is more of a circular dependency with algebraic geometry than a hard one. It's good to have walking in, but realistically you can't master the subject without knowing algebraic geometry.
You'll also need analysis, in particular complex analysis for curves. Real analysis and topology should also be covered but I suppose with tenacity you could get by without them.
To translate these into concrete suggestions, in your position I'd try to work through the following, in order:
1. Linear Algebra Done Right (Axler)
2. Abstract Algebra (Dummit & Foot)
3. Complex Analysis (Ahlfors)
4. Algebraic Curves (Fulton)
The last one is a standard upper undergraduate introduction to the subject.
If possible you should organize a study group or take a class though, because trying to learn math on your own from a textbook is rough.
Edit: Here is another overview that seems good:
A cross between a wiki and a collaborative textbook.
So I think understanding Weil conjectures are key for modern algebraic geometry. And it's always easier to understand algebraic curves (algebraic geometry with dimension 1) and their connection to Riemann surfaces (algebraic curves over the complex numbers with analytic rather then algebraic structure), as they provide motivation for many of the results and constructions.
A good introduction to Algebraic Curves and the Weil conjectures I've found is following
For general algebraic geometry, JS Milne's notes are rather good
and for an introduction to commutative algebra Atiyah-Macdonald's book is great.
Igor R. Shafarevich, Basic Algebraic Geometry, two volumes, third edition, Springer, 2013.
Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, 1994. (Especially nice if you like complex analysis, differential geometry and de Rham theory.)
David Eisenbud and Joseph Harris, The Geometry of Schemes, Springer, 2010. Currently at https://www.maths.ed.ac.uk/~v1ranick/papers/eisenbudharris.p...
But whenever I want to learn some new field of math I start with Wikipedia. This is a pretty good overview:
I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!…
A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet finally it surrounds the resistant substance.
— A. Grothendieck
"Ideals, Varieties, and Algorithms" is an "invitation to computational geometry" by Cox, Little, and O'Shea. I think parts of it might really appeal to HN readers and it's supposed to be for an undergrad math major audience.
As they say, a variety is the zero set of some polynomial equations. In particular a Riemann surface can be represented as a polynomial equation P(x,y)=0. This can quantised as you might expect x->x, y-> h d/dx, and you can study the equation P(x,h d/dx) \psi(x) = 0. The solution \psi can be written as a formal power series. For a nice choice of P(x,y), these power series can be a generating function for "something", and can contain a lot of interesting algebraic information.
Hopefully, someday when we see Joan Baez represented in movies, we’ll see her physicist father, perhaps giving a physics book to his 8 year old nephew.