\part{Fundamentals} Mathematical logic Set theory Group theory Order theory Real analysis Ordinary differetial equations Linear algebra General topology Vector analysis Fourier analysis Partial differential equations Differential topology Representation theory? \chapter{Lie groups} Lie theory is often explained as being an analogue of Galois theory for differential equations. Galois theory was motivated by the idea of using transformation groups to study polynomial zeros. More concretely, given a polynomial $P$, we want some transfomation group $G$ (with an action) such that with some $r$ satisfying $P(r)=0$ and $g\in G$, one has $P(\alpha(g,r))=0$; the transformation group tells us how to map zeros to other zeros. This perspective gives interesting methods to recover zeros of polynomials, but it also permits proofs of the Abel-Ruffini theorem which states that there is no general formula in radicals for polynomials of degree 5 and above. Lie theory attempts to create a similar model for solutions of differential equations rather than solutions of polynomial equations. We seek a transfomation group $G$ (with an action) such that with some $y$ satisfying $f(x,y,y',\hdots)=0$ and some $g\in G$, we have $f(\alpha(g,(x,u,u',\hdots)))=0, u=\alpha(g,y)$. Though the spirit of both theories are extremely close, the differing nature of the equations and the nature of the symmetries permit theories of very different flavors. One of the main differences is that Lie theory describe the ways in which a solution may smoothly flow toward other solutions, while Galois theory describes the technique was to describe transformation groups that map any zero offor polynomials (i.e groups that when acting on a polynomial's domain map zeros of the polynomial to other zeros). Lie theory similarly tries to study differential equations by means of symmetry groups, however due to the differing nature of differential equations, one requires the use of "continuous" (technically "smooth") symmetries to develop a successful theory. Unfortunately this definition of a Lie group offers practically no insight whatsoever as to why we want to consider them \section{Continuous transformation groups} Since solutions of differential equations are functions on a differentiable manifold, the theory is developed in the context of differential topology, where the field is applied and enriched by the fields of differential geometry and differential equations. Given a "structure" $X$, an \emph{automorphism group} is a group $G$ of the automorphisms of $X$; these automorphisms naturally give a way for $G$ to act on $X$. Most of the time there are so many different automorphisms with varying structure, so we often consider subgroups of this group; subgroups of automorphism groups are \emph{transformation groups}. Advanced group theory is used to model symmetries on finite spaces, Transformation groups appear and are studied in various fields of mathematics, depending on what types of spaces the transformation group is modelling (polynomial zeros, linear transforms on shapes that leave them invariant, solutions of a differential equation) and "how" the transformation group functions (continuous, discrete, or smooth symmetries) \begin{definition}[Topological group] \end{definition} To model continuous symmetries we require not only a group, but one that can "act smoothly". This desire is satisfied by employing \emph{Lie groups} as continuous transform groups; they are groups that are also smooth manifolds with smooth operations. \section{Lie groups} The titular object that Lie theory concerns itself with is \emph{Lie groups}; smooth manifolds that are also a group. When used as continuous transformation groups, they are particularly nice to work with since their topology is very nice. \begin{definition}[Lie group] Lie group is a group $(G,\cdot)$ such that $G$ is also a smooth manifold and multiplication and the inverse operation are smooth functions. \end{definition} Say we have an $n$-order PDE, we consider some submanifold of $X \times U^{(n)}$ representing solutions of said PDE; by studying the vector fields on this submanifold (with the help of a tool called a "prolongation") we can determine a basis to represent any vector field on this submanifold, then finally "exponentiate" it to calculate the orbits of the symmetry group. \begin{definition}[Lie subgroups] \end{definition} Closed-subgroup theorem Let $G$ be a Lie group, then any closed subgroup $H$ is an embedded Lie group. \section{Transformation groups} Transformation groups are groups of "transformations" (or in the worst cases, just elements that correspond to a transformation) that act on some space. Most Lie groups we will see are groups formed by special automorphisms that act on the manifold; these are known as \emph{transformation groups}. Transformation groups are associated with a Lie group action semiregular actions are those whose orbits are embedded smooth manifolds that all have the same dimension. Alternatively, they are such that the stabilizer of each point on manifold is trivial. \section{Local Lie group} One often only wants to deal with an open neighborhood of the identity, and therefore end up formally working with a \emph{local Lie group} instead of a (global) Lie group. While our definition for a local Lie groups removes the guarante of the closure of the group operation, it does allow working directly with some local coordinates of a chart covering said this neighborhood. \begin{definition}[Local Lie group] Essentially a Lie Pregroup \end{definition} Local lie groups are always a neighborhood of some global Lie group \begin{proposition} Let $(G,\cdot)$ be a connected Lie group and $U$ be an open neighborhood of $1_G$, then any element $g\in G$ is a finite product of elements in $U$ \end{proposition} \subsection{Local transformation groups} There is a local analogue of transformation groups too. \section{Classical Lie groups} The matrix groups are perhaps the most frequenly occuring examples of Lie groups. In many contexts they are used as transformation groups for some As a manifold, $\mathrm{GL}(n)$ is interpreted as the subspace topology of the Euclidean topology $\mathbb{R}^{n^2}$; this topology immediately makes matrix groups Lie groups.