
Course Notes – J.S. Milne - lolptdr
http://www.jmilne.org/math/CourseNotes/index.html
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tw1010
The real niche that needs to be filled here is not more textbooks, it is a
really really good description of how to actually teach yourself this stuff on
your own. There are so many ways to get this wrong. It's insanely easy to
trick yourself into thinking you understand these things without actually
doing so. I've read some of the stuff Thurston and Grothendieck and people on
HN have written about this, but it still doesn't feel enough. Just doing
exersizes is not enough when you get to the graduate level, at that point
introspection and the inner mental game becomes much more significant. And
without the collaborative help from a greater wiser community everyone is left
to discover this stuff on their own, which means a lot of people are going to
fail when other interests take priority.

~~~
aphextron
>The real niche that needs to be filled here is not more textbooks, it is a
really really good description of how to actually teach yourself this stuff on
your own.

I don't think you can. Sure there are a select few insanely motivated geniuses
who could do it. But learning math is just really, _really_ hard. Harder than
anything most people ever come in contact with. It's kind of like being able
to punch yourself in the face. You can try to, but you'll never be able to do
it with the same force as someone else. Or like being on a sports team, where
a good coach will push you far beyond the limits of what you could have
possibly set for yourself. This is coming from someone who taught themselves
to code completely solo from nothing and built a successful career as a
software engineer. I've recently gone back to school because I simply could
not teach myself even remedial high school math with sufficient rigor to
prepare for upper division science/engineering courses.

~~~
GregBuchholz
>learning math is just really, really hard.

I'll accept the premise, but I still wonder if there are things that can be
done to make it easier for someone. In my case, I've been trying to learn some
more mathematics recently, and one of the most annoying things is coming
across notation that isn't defined in a paper, presumably because "everyone"
who can read the paper is familiar with the context and knows what the "skinny
long arrow" means (good luck with that internet search). I wonder if there
could be a wiki-like / forum / stackoverflowish site, which people could use
to discuss and provide running commentary on a paper/book. Especially useful
would be the ability for people to be able to annotate the paper by
translating the formulas in to a formal language where you could track down
the definition of the various operators, and try to figure out why the author
used both of → and ↦ in the paper, when they both appear to be for
functions/maps. (Just to preempt the easy objections, I'm not trying to
suggest that each paper be formalized and proven in something like
Isabelle/Coq).

In the ideal form, this website would allow you to see the paper or book page
in question, and then see all the people who commented or had questions on
each particular sentence (in the margin?). There could be filtering and voting
so that experts could bypass the newbie commentary, etc..

I suppose part of my problem would be solved by getting a book like:

[https://www.amazon.com/Mathematical-Notation-Guide-
Engineers...](https://www.amazon.com/Mathematical-Notation-Guide-Engineers-
Scientists/dp/1466230525/ref=sr_1_1?ie=UTF8&)

...(which I just came across when composing this message).

Maybe someone has a other suggestions for something like this? Maybe a site
similar to this already exists?

And on a slightly related note to making things easier to learn, I think
learning programming is much easier than math, because even though both are
abstract, at least with programming you get a tangible, concrete thing (the
program) that you can run and modify and extend, and the computer will tell
you when you went wrong (e.g. won't compile, output result is unexpected,
etc.).

~~~
sn9
Forgive me if I'm making incorrect assumptions about your background, but
usually you learn math from books of varying degrees of difficulty which
naturally force you to become accustomed to various kinds of notational
conventions.

You wouldn't try to learn math from papers until you've built that foundation
(unless you have access to a tutor/mentor), at which point the notation
usually shouldn't be an issue.

~~~
GregBuchholz
That sounds like the traditional method of learning math. I was wondering if
we could leverage technology and our experiences with teaching/learning the
formal systems of programming languages to make more math more accessable. For
instance, I'm thinking this little instance of geometric algebra:

[http://www.shapeoperator.com/2016/12/12/sunset-
geometry/](http://www.shapeoperator.com/2016/12/12/sunset-geometry/)

...might be easier for me to understand if I could use Haskell to implement
the wedge and geometric product operators on an algebraic data type describing
the scalar/vector/bi-vector thingy. There is probably an applied vs. pure
thing here as well. My motivations for investigating geometric algebra is to
see if geometric algebra makes synthesizing mechanical linkages easier,
whereas maybe most expositions on geometric algebra are focused on teaching
geometric algebra to advance the state of geometric algebra. That's probably a
long winded way of saying that mathematicans are writing for mathematicians
(whether by design or accident). I suppose I should re-read Mindstorms again,
but this time in the context of adult learning.

~~~
sn9
I'm not sure if this is what you're looking for, but I've had this book on my
wishlist for a quite a while and it seems to fit:
[http://www.geometricalgebra.net/](http://www.geometricalgebra.net/)

~~~
GregBuchholz
Yes, that looks to be exactly the type of thing I'm thinking of. Thanks.

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stablemap
His book on linear algebraic groups finally comes out via CUP in a couple of
weeks. I haven't read recent drafts but it should address a real need:

[http://www.jmilne.org/math/Books/iag.html](http://www.jmilne.org/math/Books/iag.html)

I've found his notes most useful when, because of the medium, he is able to
provide more background, detail and examples compared to the standard texts.

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fhood
I read the first chapter on algebraic group theory, and was shocked at how
unhelpful it would be to an undergrad attempting to learn the subject. Then I
realized I was a dolt, and read some of the Group theory chapter which did, in
fact, effectively explain the terms and concepts thrown about in the AGT
chapter. Fair enough, but I prefer my college textbook which merged the two.

~~~
stablemap
Most of these texts were originally intended for graduate courses at Michigan.
In any event, I think it's helpful to note that algebraic groups are to
algebraic geometry as Lie groups are to differential geometry; it's not a
primary subject, even if the name does not emphasize this.

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jordigh
Ooh, nice a discussion of complex multiplication. I first heard about this a
few weeks ago during the Mathematical Congress of the Americas, and I was a
bit embarrassed to admit (to a group of number theorists) that I had never
even heard about this. They were happy to explain the basic idea, although I
still need a lot of time to digest the mechanics of it.

~~~
stablemap
It's a third year course! If you want a more leisurely treatment I hear good
things about Cox's book "Primes of the Form x²+ny²". Even if you don't reach
the CM part you will have gained a lot of motivation for it, all the way back
to _Disquisitiones_.

~~~
jordigh
Ooh, that's related? I've read through parts of Cox but hadn't gotten deep
enough to realise that he does CM! Thanks for the hint!

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mercurystills
Thanks for sharing. Good stuff.

