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How I Learned to Love Algebraic Geometry (johncarlosbaez.wordpress.com)
243 points by chmaynard 7 days ago | hide | past | web | favorite | 54 comments





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 geometry: https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...


More than over any field, over any ring. For example, that allows you to look at families of curves or talk about reduction modulo p (if you do number theory) very nicely/naturally.

A mathoverflow thread with cool uses of schemes: https://mathoverflow.net/questions/59071/what-elementary-pro...


The book by Gelfand, Kapranov and Zelevinsky is a magical mystery tour of precisely this.

I liked the part about Alexander This one: One of these geniuses was Hartshorne’s thesis advisor, Alexander Grothendieck. From about 1960 to 1970, Grothendieck revolutionized algebraic geometry as part of an epic quest to prove some conjectures about number theory, the Weil Conjectures. He had the idea that these could be translated into questions about geometry and settled that way. But making this idea precise required a huge amount of work. To carry it out, he started a seminar. He gave talks almost every day, and enlisted the help of some of the best mathematicians in Paris.

Indeed. Grothendieck is one of the few names I associate with the word "genius" without qualification. Up there with people like von Neumann and Witten. There are a lot of really smart people in the world, but they stand out even among the best.

I really liked his first paragraphs! I also liked physics and was a math grad student to learn the math for physics!

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.


“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?

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 polynomials! And all closed projective complex manifolds are algebraic varieties. See https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...


> 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 transformation, sines and cosines would be very natural operations that would have a deep generelization when sufficiently abstracted.


The Fourier transform is just a change of basis, if you squint at it properly. So at least some of the time (eg, when you're not mixing frequency and time domains like a crazy person) you can convert to frequency space and the functions become much simpler, and usual polynomial operations are relevant again.

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 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.


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.


There are many other tools for studying these pathological cases: symplectic geometry, different flavors of algebraic geometry that look at orbifolds (manifolds with corners), schemes (lots of nice singularities), deformation theory... There's a fun paper that I think of as the "bad as you wanna be" paper, about deformations of various mathematical objects: https://arxiv.org/abs/math/0411469

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.


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.

What if the approach to the simpler cases is not applicable to the harder ones. Studying the simple may shape ones thoughts in a way that prevents seeing the way to harder problems. Maybe? Sometimes new things are discovered because someone didn't know any better and tried something they were not ready for. Innovation sometimes comes from the self taught.

Complex numbers is one area where sine is very relevant.

Fascinating article about the geometry of complex numbers: https://acko.net/blog/how-to-fold-a-julia-fractal/


Can geometric algebra be used in algebraic geometry? I understand geometric algebra is useful in physics involving differential geometry and calculus, and that algebraic geometry is heavy on the algebra of polynomials and a complete field of study in mathematics, whereas geometric algebra is more like an object (Clifford algebra, etc.).

Geometric algebra is about the algebra of a widget called a grassmanian, which figures very prominently in algebraic geometry.

I have been playing with John Browne's Grassmann Algebra in Mathetmatica for my geometric algebra studies, but this is the first I've encountered the Grassmannian. Thanks, I'll look into it.

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 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.


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.

That book is great, but not mainly about algebraic geometry: it's more about differential geometry.

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...

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?

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.

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:

https://www.math.ucdavis.edu/~osserman/math/pcm.pdf


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.

Take a look at the stacks project: https://stacks.math.columbia.edu

A cross between a wiki and a collaborative textbook.


The difficulty with learning 'modern' algebraic geometry is not only is it very dense and general, but that means the original motivation can become lost.

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.


Depends how much math you know, but Chapter 15 (Commutative Rings and Algebraic Geometry) of Dummit and Foote may be a good starting place.

My favorite intros to algebraic geometry are these:

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:

https://en.wikipedia.org/wiki/Algebraic_geometry


I highly recommend the book "An Invitation to Algebraic Geometry" by Karen E. Smith et. al., as an introduction. It'll give you a taste of the field, without delving too deeply into technicalities.

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)

Vakil's notes have an interesting epigram from Grothendieck:

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


Vakil seems very fond of that quote, the full title of his notes is "The Rising Sea - Foundations of Algebraic Geometry"!

If you do have some math background, "An Invitation to Algebraic Geometry" by Karen Smith et al is a lovely, slim volume that gets to important ideas right away.

"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.


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!

Try some projective geometry to get your feet wet.

It would be nice if they went in to a bit more detail about what they see as the connection between these two subjects.

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.


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.


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.

Cool! It sounds fun. Back then she used to sing at the so-called Nameless Coffeehouse in Harvard. Unfortunately I was 7 years old in 1968, so I missed all this.

He was also the crew-cut guy running demos in film loops I saw in school when I was a kid in the '60s and '70s.

He’s written several. Me and my grad student colleagues found “Gauge fields, knots and gravity” to be very accessible

I wrote "Gauge Fields, Knots and Gravity", but we were talking about my uncle, who wrote "The New College Physics: a Spiral Approach".

Oh sorry, I thought they were talking about you too

oh I never realized they were siblings .. https://ncatlab.org/johnbaez/show/HomePage

they aren’t. joan and john are cousins.

vocabulary fail on my part, I thought siblings was a larger class

Algebraic Geometry, not to be confused with Geometric Algebra [1]! I guess naming things was never the strong side of mathematicians...

[1] https://news.ycombinator.com/item?id=13239632


I like that you can use latex in the comments section..



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