Commit 472a4391 by 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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