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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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