
The Geometry of Reflection Groups (2015) [pdf] - espeed
http://people.mpim-bonn.mpg.de/geordie/mpg.pdf
======
KevinSorboFan
Coxeter groups are fun. I took it as a graduate course, but honestly you can
get pretty far (and we did) just using elementary proof techniques that a 1st
or 2nd year undergrad could handle.

I would recommend the book by Michael Davis for an in-depth dive:
[https://people.math.osu.edu/davis.12/davisbook.pdf](https://people.math.osu.edu/davis.12/davisbook.pdf)

~~~
JadeNB
> I would recommend the book by Michael Davis for an in-depth dive:
> [https://people.math.osu.edu/davis.12/davisbook.pdf](https://people.math.osu.edu/davis.12/davisbook.pdf)

While it's a great book, I think it's very intimidating for a newcomer to the
field. (I assume, without meaning anything pejorative by it, that a lot of the
HN interest in mathematics comes from amateur mathematicians with varying
backgrounds.) Humphreys ([https://www.cambridge.org/core/books/reflection-
groups-and-c...](https://www.cambridge.org/core/books/reflection-groups-and-
coxeter-groups/2910C1E00877D33A04A512791B6EDD72)) or Grove and Benson
([https://www.springer.com/us/book/9780387960821](https://www.springer.com/us/book/9780387960821))
is probably a better introduction.

------
bgilroy26
Understanding Lie groups and differential equations is my current life goal,
but I only took math up to linear algebra and differential equations for
scientists and engineers in undergrad.

This is very readable, thank you for sharing!

~~~
romwell
I think you are almost there in terms of being prepared; I'd recommend
studying the following before tackling Lie groups (if you haven't yet):

-multivariate calculus (any text will do)

-differential geometry (Manfredo DoCarmo's diff. geom. in 3D)

-introduction to topology (Munkres)

Multivariate calculus gives you the language to tackle surfaces in terms of
their shape, which is what differential geometry studies.

Topology gives you more language to study surfaces in terms of their
structure, and shows you the connection between structure and algebra.

Lie Groups/Algebras connect the shape and algebra, and so need differential
geometry, and benefit from understanding of topology.

~~~
JadeNB
Also, as with a lot of places where algebra meets geometry, the 'homogenising'
(in a good sense) effect of the algebra makes the geometry much less
challenging than it could otherwise be. (Linear algebraic groups, which are a
particular kind of scheme, underlie much of my research; but I understand way
less about schemes in general than about the geometric structure of LAG's.)
That's just to say that, from my point of view (as, admittedly, an
algebraist), getting the solid linear-algebra background to handle Lie
algebras as abstract structures is way more important to understanding Lie
groups than a full picture of all the subtleties of differential geometry.

(With that said, anyone who wants to understand the geometry of differential
equations _will_ need at some point to handle a lot of non-homogeneous
geometry, so my advice isn't the end of the story even for those who agree
with it.)

~~~
romwell
>getting the solid linear-algebra background

Oh! What an omission I made.

A solid linear algebra background is a must for absolutely _everything_.

There's nothing one can do without it. Not Calculus in several variables, not
geometry, not group theory, etc.

I sort of assumed that people with diff. eq. under their belt know linear
algebra well, but it's often not the case.

I can strongly recommend the following two texts for the subject:

-Sheldon Axler: Linear Algebra Done Right

-Serge Treil: Linear Algebra Done Wrong

