Commit 472a4391 authored by Jannes Bantje's avatar Jannes Bantje

start work on subsection

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......@@ -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.}
% section index_theory (end)
% chapter elliptic_genera_phd_seminar (end)
\ No newline at end of file
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