@@ -407,6 +407,32 @@ This gives a ring homomorphism

\end{itemize}

\end{remark}

\subsection*{Elliptic Operators and Elliptic Complexes}

In the following everything is considered to be smooth, this applies in particular to manifolds, vector bundles\footnote{meaning, that they are smooth manifolds} over them and their sections.

We now start define the main players of index theory, namely differential operators:

Let $X$ be a closed manifold and $E,F$ complex smooth vector bundles over $X$.

\begin{definition}

A linear map $D \colon\Gamma(E)\to\Gamma(F)$ is called \Index{differential operator} of order $p$, if

\begin{enumerate}[(i)]

\item$D$ is local, i.e. if $s \in\Gamma(E)$ vanishes on an open subset, then $D s$ does as well,

\item in local coordinates

\[

D s =\sum_{\abs*{\alpha}\le p} A^\alpha\diff{^\alpha}{x_\alpha}

\]

where $A^\alpha$ is a smooth matrix valued function.

\end{enumerate}

\end{definition}

(see also \cref{def:diff_op-order-one,def:diff_op_general})

\begin{remark}

\begin{itemize}

\item First order differential operators suffice for many applications, for example \textcite{higson_roe} contend themselves with first order differential operators.

\end{itemize}

\end{remark}

\todo[inline]{\textcite[Thm.~III.13.13]{lawson_spin} is the closest thing to the Atiyah--Singer formula in \cite{hirzebruch_modularforms} I found so far.}