If, like me, you're not a real mathematician but suffered through linear algebra and differential equations, you can still totally understand this stuff! I started off teaching myself differential geometry but ultimately had far more success with lie theory from a matrix groups perspective. I highly recommend:
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!
My experience with groups and linear algebra is similar. I made real progress only after I got past the initial fear and intimidation, making a point of understanding those beautiful equations. Now I find myself agreeing with those who argue that mathematics education could profitably begin with sets and groups instead of numbers.
Second this! And if you want a part memoir part history of this subject as it relates to physics (through Langlands Program) part ode to the beauty of maths, I recommend reading Edward Frenkel's Love & 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.
What I always miss from this introductory abridged explanations, and what makes the connection between Lie groups and algebras ('infinitesimal' groups) really useful, is that the exponential process is a universal mechanism, and provides a natural way to find representations and operators (eg Lie commutator, the BCH formula) where the group elements can be transformed through algebraic manipulations and vice-versa. That discovery offers a unified treatment of concepts in number theory, differential geometry, operator theory, quantum theory and beyond.
> For instance, the fact that the laws of physics are the same today as they were yesterday and will be tomorrow — a symmetry known as time translation symmetry, represented by the Lie group consisting of the real numbers — implies that the universe’s energy must be conserved, and vice versa. “I think, even now, it’s a very surprising result,” Alekseev said.
Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that weren’t the case?
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.
1. Lie groups describe local symmetries. Nothing about the global system
2. From a SR point of view, energy in one reference frame does not have to match energy in another reference frame. Just that in each of those reference frames, the energy is conserved.
3. The conservation/constraint in GR is not energy but the divergence of the stress-energy tensor. The "lost" energy of the photo goes into other elements of the tensor.
4. You can get some global conservations when space time exhibits global symmetries. This doesn't apply to an expanding universe. This does apply to non rotating, non charged black holes. Local symmetries still hold.
That symmetries imply conservation laws is pretty fascinating (see the Noether theorem). I guess it seems only strange it you assume already that the conservation law holds.
It is surprising that you can derive conversation laws entirely from the symmetry of lie groups, and that every conservation law can be tied to a symmetry.
Such a bad (AI written?) article. These kind of introduction to advanced topics feels like how to draw an owl tutorial where they spent so much time diving into what group is.
> 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.
The quality of this article is par for the course for Quanta Magazine, sadly. I do not need to accuse the author of using AI to explain the data I'm seeing here. It feels like every submission on HN from Quanta garners the exact same discussion: The article is almost worthless because it presents complex ideas in such a cheap, dumbed-down, and imprecise way that it ceases to communicate anything interesting. (Interested readers can fare much better by reading other sources.) It's been this way for years. The phenomenon is almost Wolfram-Derangement-Syndrome-like.
That is very strange. It's certainly not an academic level explanation, but that's not what the magazine is for. But the blatant incorrect statement is beyond the pale. Dim(SO(N)) = N(N-1)/2. Thus SO(4) has dimension 6.
I hate statements like this due to their imprecision and their contribution to making mathematics difficult to learn.
> 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.
If, like me, you're not a real mathematician but suffered through linear algebra and differential equations, you can still totally understand this stuff! I started off teaching myself differential geometry but ultimately had far more success with lie theory from a matrix groups perspective. I highly recommend:
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!
My experience with groups and linear algebra is similar. I made real progress only after I got past the initial fear and intimidation, making a point of understanding those beautiful equations. Now I find myself agreeing with those who argue that mathematics education could profitably begin with sets and groups instead of numbers.
https://d1gesto.blogspot.com/2025/11/math-education-what-if-...
Second this! And if you want a part memoir part history of this subject as it relates to physics (through Langlands Program) part ode to the beauty of maths, I recommend reading Edward Frenkel's Love & Math:
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.
What I always miss from this introductory abridged explanations, and what makes the connection between Lie groups and algebras ('infinitesimal' groups) really useful, is that the exponential process is a universal mechanism, and provides a natural way to find representations and operators (eg Lie commutator, the BCH formula) where the group elements can be transformed through algebraic manipulations and vice-versa. That discovery offers a unified treatment of concepts in number theory, differential geometry, operator theory, quantum theory and beyond.
> For instance, the fact that the laws of physics are the same today as they were yesterday and will be tomorrow — a symmetry known as time translation symmetry, represented by the Lie group consisting of the real numbers — implies that the universe’s energy must be conserved, and vice versa. “I think, even now, it’s a very surprising result,” Alekseev said.
Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that weren’t the case?
It's funny you say that, because energy actually isn't conserved in general.
One somewhat trivial example is that light loses energy due to redshift since photon energy is proportional to frequency.
Where does the energy go then?
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.
There are certain answers to the above question
1. Lie groups describe local symmetries. Nothing about the global system
2. From a SR point of view, energy in one reference frame does not have to match energy in another reference frame. Just that in each of those reference frames, the energy is conserved.
3. The conservation/constraint in GR is not energy but the divergence of the stress-energy tensor. The "lost" energy of the photo goes into other elements of the tensor.
4. You can get some global conservations when space time exhibits global symmetries. This doesn't apply to an expanding universe. This does apply to non rotating, non charged black holes. Local symmetries still hold.
That symmetries imply conservation laws is pretty fascinating (see the Noether theorem). I guess it seems only strange it you assume already that the conservation law holds.
It is surprising that you can derive conversation laws entirely from the symmetry of lie groups, and that every conservation law can be tied to a symmetry.
We are running a live online bootcamp, Group Theory 360: https://quantumformalism.academy/group-theory-360.
Lie groups are central part of the bootcamp where we will cover their applications beyond physics including geometric deep learning!
Such a bad (AI written?) article. These kind of introduction to advanced topics feels like how to draw an owl tutorial where they spent so much time diving into what group is.
> 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.
The quality of this article is par for the course for Quanta Magazine, sadly. I do not need to accuse the author of using AI to explain the data I'm seeing here. It feels like every submission on HN from Quanta garners the exact same discussion: The article is almost worthless because it presents complex ideas in such a cheap, dumbed-down, and imprecise way that it ceases to communicate anything interesting. (Interested readers can fare much better by reading other sources.) It's been this way for years. The phenomenon is almost Wolfram-Derangement-Syndrome-like.
That is very strange. It's certainly not an academic level explanation, but that's not what the magazine is for. But the blatant incorrect statement is beyond the pale. Dim(SO(N)) = N(N-1)/2. Thus SO(4) has dimension 6.
The “tangle of spheres and circles” is probably a reference to the Hopf fibration.
Correct. I have all of this worked out if anyone wants to check my work. I validated it through John Baez.
I hate statements like this due to their imprecision and their contribution to making mathematics difficult to learn.
> 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.