A Norwegian mathematician named Sophus Lie was looking at differential equations and
wanted to solve them with a single, unified theory.
His idea was to use symmetries.
Why were symmetries in his head?
He was inspired by an earlier mathematician Galois, who was studying polynomial equations.
His idea was also symmetries and Galois theory was extremely successful.
Lie wanted to replicate the success of Galois theory and made the analogy between differential
equations, and polynomial equations.
Now the symmetries in Galois theory were permutations of the roots of the polynomial, which are
“discrete” symmetries.
“Discrete” in the sense that there are finitely many roots of a polynomial.
However, the symmetries associated with differential equations would be continuous.
That’s because for differential equations, there is in general a continuous family of
solutions, and it is only until you specify the initial conditions that you have one single
solution.
So you can say that in this illustration, there is a vertical translational symmetry
in the set of solutions.
And the symmetries here are continuous as you can continuously transform from one solution
to another.
So this is the analogy that Lie was going for.
Unfortunately, while Galois theory became the dominant tool to study polynomial equations,
Lie’s theory of continuous symmetries did not dominate the study of differential equations.
However, the study of continuous symmetries still proves very useful, just not necessarily
differential equations.
For example, a lot of physical systems exhibit symmetries.
The physical laws do not change when you rotate around the origin.
If we apply this idea of symmetry to quantum systems, somehow we end up with the weird
concept of spin 1/2.
But perhaps more surprisingly, the study of symmetries predicted the existence of the
omega baryon, experimentally confirmed a few years later.
Here are just more physical applications, which I am more familiar with, but there are
plenty of applications in more advanced areas in mathematics as well.
So while Lie theory started from trying to understand differential equations, it turns
out to be useful in a lot of other areas as well.
For this, I am going to make a video series on the theory of Lie, and as of now, the plan
is 8 videos.
This is already the first video talking about why we need to care about continuous symmetries.
Then before giving a high-level overview of the theory, there is just 1 video of prerequisites
about group theory, specifically real and complex rotational symmetries.
Because the exponential map is particularly important in the theory, I want to spend a
video on more general exponential maps, including on vectors and differential operators.
Next, two relatively more standard videos on Lie algebras and Lie brackets, and how
it makes our lives easier.
Finally, two videos on quantum spin, from a more mathematical perspective.
So that’s the plan for the whole series.
Maybe I will tweak it here and there, but that’s the plan for now.
And I hope I will see you in the next video.
As usual, if you like this video, like, subscribe, comment, and share.
It will help a lot.
See you next time!