Re:How I Learned to Love Algebraic Geometry
by Grustaf on Saturday March 16, 2019 @01:43PM
(#19409063)
Everyone should also read Modern Geometry book by Dubrovin, Novikov and Fomenko, it starts pretty basic and covers a lot of ground, and is extremely well written.
Re:How I Learned to Love Algebraic Geometry
by johncarlosbaez on Sunday March 17, 2019 @03:07PM
(#19415767)
That book is great, but not mainly about algebraic geometry: it's more about differential geometry.
Re:How I Learned to Love Algebraic Geometry
by Grustaf on Sunday March 17, 2019 @03:36PM
(#19416002)
I know, but itäs an excellent introduction to geometry in general, like some of the other books listed. It’s probably not even possible to get from 0 to algebraic geometry in a single book...
Re:How I Learned to Love Algebraic Geometry
by NiftyWalrus on Saturday March 16, 2019 @09:28AM
(#19407715)
Could you describe what Grothendieck was doing in algerbiac geometry? Would studying the above get you up to the point of his work? If not, what are concrete suggestions that would get you there?
Re:How I Learned to Love Algebraic Geometry
by obastani on Saturday March 16, 2019 @10:21AM
(#19407925)
The "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution:
Re:How I Learned to Love Algebraic Geometry
by NiftyWalrus on Saturday March 16, 2019 @10:37AM
(#19407993)
What is the modern path to study this area now? I’m sure it’s better understood now and one wouldn’t have to follow the historical approach to study the same concepts.
Re:How I Learned to Love Algebraic Geometry
by obastani on Saturday March 16, 2019 @10:54AM
(#19408099)
I don't know myself, but here are some possible answers:
https://mathoverflow.net/questions/114034/learning-path-for-...
Edit: Here is another overview that seems good:
Re:How I Learned to Love Algebraic Geometry
by sdenton4 on Saturday March 16, 2019 @03:18PM
(#19409614)
Yeah, it's still incredibly difficult terrain, despite all the time that's passed. Wrestling with Hartsthorne is still a rite of passage for students in this area. I am (very) intrigued to know what would happen if someone made a serious effort to make the ideas more accessible. It remains an area where the depth of knowledge required is legitimately deep: would be cool to figure out which ideas are actually independent of other parts of the stack.
Re:How I Learned to Love Algebraic Geometry
by MaysonL on Saturday March 16, 2019 @11:11AM
(#19408200)
Take a look at the stacks project: https://stacks.math.columbia.edu
A cross between a wiki and a collaborative textbook.
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.