
How I Learned to Love Algebraic Geometry - chmaynard
https://johncarlosbaez.wordpress.com/2019/03/15/algebraic-geometry/
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kevinventullo
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...](https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry)

~~~
heinrichf
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...](https://mathoverflow.net/questions/59071/what-elementary-problems-can-
you-solve-with-schemes?answertab=votes#tab-top)

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graycat
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.

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umutisik
“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?

~~~
kevinventullo
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...](https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry)

~~~
wolfgke
> 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.

~~~
sdenton4
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.

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eggy
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.).

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

~~~
eggy
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.

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maaaats
What are some good resources for grasping or learning more about algebraic
geometry?

~~~
throwawaymath
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.

~~~
Grustaf
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.

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

~~~
Grustaf
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...

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foxes
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.

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melling
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.

~~~
chmaynard
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.

~~~
johncarlosbaez
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.

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rsp1984
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](https://news.ycombinator.com/item?id=13239632)

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

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cpone10
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.

~~~
throwawaymath
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.

