Commit 95f9307f authored by Jannes Bantje's avatar Jannes Bantje

more outline

parent 29b73cbe
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......@@ -160,15 +160,34 @@ We therefore have
It was proven by \textcite{hirzebruch_Lgenus}, that this genus coincides with the signature.
\end{example}
Most of the genus, that are of particular interest, belong to the following class:
\begin{definition}
A genus $\varphi$ is called an \Index[genus!elliptic]{elliptic genus}, if the corresponding power seires $f(x)=x/Q(x)$ satisfies on of the three equivalent conditions:
\begin{align}
\begin{align*}
(f')^2 &= 1- 2 \delta \cdot f^2 + \varepsilon \cdot f^4 \\
f(u+v) &= \frac{f(u)f'(v) + f'(u)f(v)}{1- \varepsilon \cdot f(u)^2 f(v)^2} \\
f(2u) &= \frac{2f(u)f'(u)}{1-\varepsilon \cdot f(u)^4}
\end{align}
\end{align*}
\end{definition}
\begin{remark}
\todo[inline]{add motivation}
\end{remark}
\begin{example}
The $\hat{A}$-genus is elliptic.\todo[inline]{extend}
The $\hat{A}$-genus can be generalised to the spin bordism ring, where a morphism
\[
\alpha \colon \Omega^{\Spin}_* \to \KO^{-*}(\pt)
\]
emerges.
This map can be represented by a map of spectra $\M \Spin \to \KO$.
\end{example}
% section introduction_by_julian (end)
\section{Ellipctic Genera -- Markus} % (fold)
......@@ -180,4 +199,10 @@ We therefore have
\label{sec:modular_forms_jens}
% section modular_form_jens (end)
\section{Formal Group Laws -- Leon} % (fold)
\label{sec:formal_group_laws_leon}
\todo{looked at chapter 3 in the book}
\todo[inline]{maybe not as important for my talk!?}
% section formal_group_laws_leon (end)
% chapter elliptic_genera_phd_seminar (end)
\ No newline at end of file
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