\section{Introduction: Genera and Multiplicative Sequences -- Julian}% (fold)

\section{Introduction: Genera and Multiplicative Sequences}% (fold)

\label{sec:introduction_julian}

\emph{This section summarises the talk by Julian extended with more details on multiplicative sequences from \cite[Chap.~1]{hirzebruch_modularforms}.}

\medskip

The story start by looking at bordism invariants.

Recall that cobordism classes of manifolds can be added by taking the disjoint union, which yields a commutative additive structure.

Taking the product yields a multiplication and together they give a graded ring.

...

...

@@ -238,9 +241,12 @@ The Stolz--Teichner program tries to construct $\tmf$ geometrically using approa

Another interesting aspect ist, that $\tmf$ arises canonically from chromatic homotopy theory.

% section introduction_by_julian (end)

\section{Ellipctic Genera -- Markus}% (fold)

\section{Ellipctic Genera}% (fold)

\label{sec:ellipctic_genera_markus}

\emph{This section contains a summary of the talk by Markus prepended with some material from \cite[Chap.~2]{hirzebruch_modularforms}.}

\medskip

We already made contact with the duplication formula

\[

\sum_{k=0}^{\infty}\varphi\enbrace*{\HP^k}y^{2k}= h(y)\qquad\text{ where }\quad h(f(x))=\frac{f(2x)}{2\,f(x)f'(x)}

...

...

@@ -279,22 +285,28 @@ The gist is, that lattices of $\mathbb{C}$ give rise to elliptic genera via thei

This involves quite a lot of computations, which we will omit here -- a very brief summary is adequate nonetheless.\todo{do it!}

% section ellipctic_genera_markus (end)

\section{Modular Forms -- Jens}% (fold)

\section{Modular Forms}% (fold)

\label{sec:modular_forms_jens}

This seems to be mostly algebra and I'm unsure, whether it is actually relevant for my talk?

\emph{This is a summary of a talk by Jens in the PhD seminar.}

\todo[inline]{short summary}

% section modular_form_jens (end)

\section{Formal Group Laws -- Leon}% (fold)

\section{Formal Group Laws}% (fold)

\label{sec:formal_group_laws_leon}

\todo{looked at chapter 3 in the book}

\todo[inline]{maybe not as important for my talk!?}

\emph{This is a summary of a talk by Leon in the PhD seminar.}

\todo[inline]{looked at chapter 3 in the book}

\todo[inline]{see also \url{http://www.map.mpim-bonn.mpg.de/Formal_group_laws_and_genera\#Examples}}

% section formal_group_laws_leon (end)

\section{Index Theory -- Jannes}% (fold)

\section{Index Theory}% (fold)

\label{sec:index_theory}

\emph{This is based on my talk in the Münster topology Phd seminar 2020.

This coverage of index theory is intended to complement \cite[Chap.~5]{hirzebruch_modularforms} by providing more background on the \enquote{classical} index theory used there.

In this writeup there are more references and details given, than in the talk -- in particular the Thom isomorphism confusion has been fixed.}

\medskip

We start with a quick recap an genera as discussed in \cref{sec:introduction_julian}.

Following \textcite{hirzebruch_modularforms} a genus is a ring homomorphism