I just wanted to bring up some discussion about an apparently essential concept for some fields in mathematics as so as for some in physics, as already mentioned in the title, I'm referring to the word "symmetry".
I'm currently studying about Lie symmetry method to search for first integrals of ODE's, and already saw that the main problem would be, exactly, to compute the symmetries, which involves a PDE.
In adition, I constantly hear "symmetry breaking" from some working with QFT. It sounds like a joke but, are we trying to find those symmetries for you guys to "break'em"?
Despite the kidding, could these two contexts be related in some degree?
By my understanding, the notion of symmetry in mathematics as well as in physics refers to the invariance of an object, a quantity, a property, etc under a certain transformation.
This might originally come from our geometrical intuitions and is reflected on day to day language: for example, we can talk about a face being perfectly symmetric when we trace a vertical line bisecting it and the face (=object) would be invariant under reflection with respect to this line (=transformation).
In QFT, symmetry breaking refers rather to the fact that, in physics, systems in general tend to evolve to low energy states. Sometimes these low energy states have less symmetries than others with higher energy, even when "the whole system is also symmetric". The classical depiction is a hill with the shape of the letter $\omega$. This hill is symmetric under reflection with respect to the middle line. We can start with a ball in the center of it, in unstable equilibrium with respect to gravity, meaning moving the ball a bit to one side will cause it to roll to the bottom of one of those $u$ shaped valleys. Therefore, the state of minimum (potential) energy (that is, these valleys) does not have the same symmetry as the state on the top or the whole hill itself, but rather lies at one side or the other of the middle line.
This is a very primitive explanation because things are actually more convoluted in practice, but it might convey the essence of the idea of symmetry breaking.