In the next two lectures we will cover the basic theory of Lie groups. Lie groups are important in many areas of mathematics (not just geometry!)—including representation theory, harmonic analysis, differential equations and more. Lie groups also crop up naturally in physics—both classically (eg. Noether's theorem that every smooth symmetry of a physical system has a corresponding conservation law), and in high-energy particle physics, via gauge theory. We'll touch upon gauge theory in Differential Geometry II when we study connections on principal bundles.

Today we proved half of the “Lie group-Lie algebra” correspondence: namely that if $G$ is a Lie group with Lie algebra $\mathfrak{g}$ then if $H \subset G$ is a Lie subgroup with Lie algebra h then $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$. In Lecture 12 we will prove the converse: if $G$ is a Lie group with Lie algebra $\mathfrak{g}$, then for any Lie subalgebra $\mathfrak{h} \subset \mathfrak{g}$, there is a unique connected Lie subgroup $H$ of $G$ whose Lie algebra is $\mathfrak{h}$.