The Geometry of Reflection Groups (2015) [pdf] | Hacker News Reader

The Geometry of Reflection Groups (2015) [pdf]

people.mpim-bonn.mpg.de

47 pointsespeed7 comments

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

> I would recommend the book by Michael Davis for an in-depth dive: 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...) or Grove and Benson (https://www.springer.com/us/book/9780387960821) is probably a better introduction.

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!

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.

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.)

>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

This is great, thank you! I have read the preface of Munkres, I'm looking forward to going through his book. He's a great writer.

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