41–50 of 56 posts
What are some good resources for grasping or learning more about algebraic geometry?
What 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 c…
Earlier 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.
What are some good resources for grasping or learning more about algebraic geometry?
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
https://math.mit.edu/~poonen/papers/curves.pdf
For general algebraic geometry, JS Milne's notes are rather good
https://www.jmilne.org/math/CourseNotes/ag.html
and for an introduction to commutative algebra Atiyah-Macdonald's book is great.
Earlier 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.
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.
What are some good resources for grasping or learning more about algebraic geometry?
"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.
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I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring. 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 polynom…
> I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring. 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 tr…
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The "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution: https://en.wikipedia.org/wiki/Weil_conjectures
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.
Algebraic Geometry is a powerful tool of number theory because much of it works over any field. It allows one to translate geometric intuition (algebraic geometry over the complex numbers) into a more algebraic environment (finite, p-adic, or number fields). 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 geom…
Joan 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.