Mathematics
Lie Groups and Their Lie Algebras
Quick fact
The Lie algebra of a Lie group is a vector space that can be seen as the 'linearization' of the group around its identity, and the exponential map lets you 'exponentiate' its elements to recover the full group locally.
Why this is interesting
You've seen symmetries like rotations of a sphere. But what if the symmetry can be applied continuously, in infinitely many small steps? How do we capture the 'tiny' rotations that build up to any rotation?