Commit 52191497 authored by Jannes Bantje's avatar Jannes Bantje

update section titles

parent 615204cf
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......@@ -3,9 +3,12 @@
\label{cha:elliptic_genera_phd_seminar}
\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
\[
......
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