These people have good intentions.
The problem is that they don't manage to go critical because there's to way for students to create content, everything is manually curated.
You can't even publicly comment on the textbooks. Or at least Ciro Santilli hasn't found a way to do so. There is just a "submit suggestion" box.
This massive lost opportunity is even shown graphically at: cnx.org/about (archive) where there is a clear separation between:Maybe this wasn't the case in their legacy website, legacy.cnx.org/content?legacy=true, but not sure, and they are retiring that now.
- "authors", who can create content
- "students", who can consume content
By Rice University.
TODO what are the books written in?
- github.com/openstax/openstax-cms Uses Wagtail CMS. So presumaby they just Wagtail's WYSIWYG.
- github.com/openstax/os-webview
Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) , that has the same properties as the group.
Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).
This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.
Or more precisely, mapping each group element to a linear map over some vector field (which can be represented by a matrix infinite dimension), in a way that respects the group operations:
As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)
- page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
- 3.6 classifies the representations of . There is only one possibility per dimension!
- 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application
Bibliography:
- www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
- www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.
en.wikipedia.org/wiki/The_Honourable_Schoolboy#Adaptations mentions:
Jonathan Powell, producer of Tinker, Tailor, Soldier, Spy (1979), said the BBC considered producing The Honourable Schoolboy but a production in South East Asia was considered prohibitively expensive and therefore the BBC instead adapted the third novel of the Karla Trilogy, Smiley's People (1979)
The best modern instrumental Western music by 
Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01Not in other sections:
- Grant Green: Idle Moments (1963)
- John Abercrombie: Timeless (1975). Just close your eyes, and imagine.
- Paco de Lucía: Almoraima (1976)
- Modern Jazz Quartet: Django (1956)
- Bill Bruford Feels Good to Me (1978). Well, with Allan Holdsworth on the g
- Jean-Luc Ponty
Some specific examples:
How to teach Explain why the subject is beautiful by Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-01 +Created 1970-01-01And if you really can't make money from a subject, there is only one other thing people crave: beauty.
You have to give the beauty motivations upfront, before boring people to death with endless prerequisites, otherwise no one will ever want to learn it.
There are two cases:
- (topological) manifolds
- differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
- Original problem posed, for topological manifolds.AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
- for differential manifolds:Counter examples are called exotic spheres.Totally unpredictable count table:is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
Take the element and apply it to itself. Then again. And so on.
In the case of a finite group, you have to eventually reach the identity element again sooner or later, giving you the order of an element of a group.
The continuous analogue for the cycle of a group are the one parameter subgroups. In the continuous case, you sometimes reach identity again and to around infinitely many times (which always happens in the finite case), but sometimes you don't.
One of the first formal proof systems. This is actually understandable!
This is Ciro Santilli-2020 definition of the foundation of mathematics (and the only one he had any patience to study at all).
TODO what are its limitations? Why were other systems created?
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