https://www.amazon.com/Lie-Groups-Introduction-Graduate-Math...
and
https://bookstore.ams.org/text-13
My friends were all putnam nerds in college and I was not, and I assumed this math was all beyond me, but once you get the linear algebra down it's great!
https://d1gesto.blogspot.com/2025/11/math-education-what-if-...
https://en.wikipedia.org/wiki/Love_and_Math
and if you went to school in maths but now have left that world, this book engenders an additional spark of nostalgia and fun due to reading about some of your professors and their (sometimes very difficult) journey in this world.
Lie groups are central part of the bootcamp where we will cover their applications beyond physics including geometric deep learning!
Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that weren’t the case?
One somewhat trivial example is that light loses energy due to redshift since photon energy is proportional to frequency.
Edit: I just looked into this & there are a few explanations for what is going on. Both general relativity & quantum mechanics are incomplete theories but there are several explanations that account for the seeming losses that seem reasonable to me.
> The group of all rotations of a ball in space, known to mathematicians as SO(3), is a six-dimensional tangle of spheres and circles.
This is wrong. It's 3D, not 6D. In fact SO(3) is simple to visualize as movement of north pole to any point on the ball + rotation along that.
> This extra property is what makes SO(2) a Lie group — it can be visualized as a smooth, continuous shape called a manifold. Other Lie groups might look like the surface of a doughnut, or a high-dimensional sphere, or something even stranger: The group of all rotations of a ball in space, known to mathematicians as SO(3), is a six-dimensional tangle of spheres and circles.
This article describes a 4D manifold (among other things):
https://en.wikipedia.org/wiki/Charts_on_SO(3)
For all I know there's a 6D one too.
> Though they’re defined by just a few rules, groups help illuminate an astonishing range of mysteries.
An astute reader at this point will go look up the definition of groups and come away completely mystified how they illuminate anything (hint: they do not).
A better statement is that many things that illuminate a wide range of mysteries form groups. By themselves, the group laws regarding these things tell you very little. It's the various individual or collective behaviors of certain groups that illuminate these areas.