memoire-m2
My M2 Memoire on mapping class groups & their representations
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\documentclass{memoires} \input{./preamble.tex} \addbibresource{references.bib} \title{Mapping Class Groups \\ \& \\ their Representations} \author{Thiago Brevidelli Garcia} \dedication{This Master's Thesis is dedicated to my dear friend Lucas Schiezari. \\ May he rest in peace.} \begin{document} \makecover \maketitle \begin{about} This is my M2 Mémoire (Master's Thesis), written in June 2024 under the supervision of Professor Maxime Wolff of the Institut of Mathématiques de Toulouse (IMT), France. The subject of the dissertation at hand is the notion of the mapping class group of a surface, their linear representations and some recent developments in the field. Namely, we discuss Korkmaz' proof of the triviality of low-dimensional representations. Throughout these notes we will follow some guiding principles. First, lengthy proofs are favored over collections of smaller lemmas. This is a deliberate effort to emphasize the relevant results. That said, this dissertation is intended to be concise. Hence numerous results are left unproved. We refer the reader to proofs in other materials when appropriate. Our main references are the beautiful book by Farb-Margalit \cite{farb-margalit}, as well as the 2023 article by Korkmaz \cite{korkmaz}. We assume the reader is already familiar with basic topology and group theory. Crucially, we \emph{do not} assume any familiarity with representation theory. Because of tight deadlines, we have opted to hand-draw all the figures. \section*{Acknowledgments} I would like to thank my friends and family for their tireless love and support. Special thanks to my girlfriend for her encouragement and understanding during this challenging year. I extend my gratitude to Professor Wolff for accepting me as his student and providing me with his invaluable guidance. I should also thank Matthieu Faitg for his help with the discussion on TQFT representations. \section*{License} This document was typeset and compiled using free software. Its \LaTeX~source code is freely available at \url{https://git.pablopie.xyz/memoire-m2}, for distribution under the terms of the \href{https://creativecommons.org/licenses/by/4.0/}{Creative Commons Attribution 4.0 license}. \end{about} \tableofcontents \input{sections/introduction} \input{sections/twists} \input{sections/presentation} \input{sections/representations} \printbibliography % Backcover \makecover[b] \end{document}