What are some good resources for grasping or learning more about algebraic geometry?
How I Learned to Love Algebraic Geometry
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Re: How I Learned to Love Algebraic Geometry
#12Joan Baez’s dad, his uncle, gave him the physics book that he wrote.
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.
Re: How I Learned to Love Algebraic Geometry
#13What are some good resources for grasping or learning more about algebraic geometry?
You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the first, I'll find out for sure next term)
Re: How I Learned to Love Algebraic Geometry
#14“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.” Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?
Solution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For example, any time you hear the word “manifold”, it refers to a space which has none of these pathologies and is entirely smooth. So the entire theory of differentiable manifolds will never encourage anything which “pinches”, or drops down a dimension, etc. On…
But aren’t we missing out on something important by glossing over these “pathological” cases?
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.
Re: How I Learned to Love Algebraic Geometry
#15What are some good resources for grasping or learning more about algebraic geometry?
Try some projective geometry to get your feet wet.
Re: How I Learned to Love Algebraic Geometry
#16What are some good resources for grasping or learning more about algebraic geometry?
Atiyah-Macdonald is a small but dense book with most of the commutative algebra you'll need to start learning algebraic geometry. You'll need to know some abstract algebra as a prerequisite (groups, rings, fields; covered in undergraduate algebra courses).
Re: How I Learned to Love Algebraic Geometry
#17Joan Baez’s dad, his uncle, gave him the physics book that he wrote. 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.
In 1968, I enjoyed taking a physics course at Harvard Summer School that was taught by Dr. Albert Baez. He had a home in Cambridge and graciously invited the entire class over for a party after the course ended. The highlight of the evening was an impromptu performance by his daughter Joan.
Re: How I Learned to Love Algebraic Geometry
#18Earlier quoted context omitted.
Solution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For example, any time you hear the word “manifold”, it refers to a space which has none of these pathologies and is entirely smooth. So the entire theory of differentiable manifolds will never encourage anything which “pinches”, or drops down a dimension, etc. On…
But aren’t we missing out on something important by glossing over these “pathological” cases? 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.
For sure we are missing something. But what hope do we have of understanding the more complicated case, before we figure out what we can know in simpler cases? The eventual goal is to move back to the more complicated cases.
Re: How I Learned to Love Algebraic Geometry
#19What are some good resources for grasping or learning more about algebraic geometry?
Atiyah-Macdonald is a small but dense book with most of the commutative algebra you'll need to start learning algebraic geometry. You'll need to know some abstract algebra as a prerequisite (groups, rings, fields; covered in undergraduate algebra courses).
A less dense book for commutative algebra is Miles Reid's Undergraduate Commutative Algebra, but it doesn't cover tensor product of modules which will definitely be needed for algebraic geometry. Also for everyone interested in Atiyah-Macdonald you MUST do the exercises, half of the book is in the exercises!
Re: How I Learned to Love Algebraic Geometry
#20What are some good resources for grasping or learning more about algebraic geometry?
This isn't a fun answer, but you basically can't without substantial background in undergrad mathematics. The subject has some of the highest prerequisites of any upper undergrad/lower grad level math course, as it touches on (and uses) just about everything you'd learn through an undergrad math degree.
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.