joshua/msc-thesis
Archived
1
Fork 0
Code Activity

Adds stub for presentation

Browse source
This commit is contained in:
Joshua Moerman 2015-01-26 17:12:42 +01:00
parent 7aa8adb2fa
commit b7d07fedb3
3 changed files with 365 additions and 0 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

20
presentation/Makefile Normal file
View file

@ -0,0 +1,20 @@
.PHONY: thesis fast dirs
# We don want to pollute the root dir, so we use a build dir
# http://tex.stackexchange.com/questions/12686/how-do-i-run-bibtex-after-using-the-output-directory-flag-with-pdflatex-when-f
thesis: dirs
xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex
xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex
cp build/Rational_Homotopy_Theory.pdf ./
fast: dirs
xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex
cp build/Rational_Homotopy_Theory.pdf ./
haltfast: dirs
xelatex -file-line-error -output-directory=build --halt-on-error Rational_Homotopy_Theory.tex
cp build/Rational_Homotopy_Theory.pdf ./
dirs:
mkdir -p build

185
presentation/Rational_Homotopy_Theory.tex Normal file
View file

@ -0,0 +1,185 @@
\documentclass[14pt]{beamer}
\definecolor{todocolor}{rgb}{1, 0.3, 0.2}
\newcommand{\td}[1]{\colorbox{todocolor}{*\footnote{TODO: #1}}}
\input{preamble}
\usepackage{tabularx}
\renewcommand{\tabularxcolumn}[1]{p{#1}}
\newcommand{\Frame}[2]{
\begin{frame}{#1}#2\end{frame}
}
\title{Rational Homotopy Theory}
\author{Joshua Moerman}
\institute[Radboud Universiteit Nijmegen]{Supervisor: Ieke Moerdijk}
\date{}
\begin{document}
\AtBeginSection[]{
\begin{frame}<beamer>
\tableofcontents[currentsection]
\end{frame}
}
\Frame{}{
\titlepage
}
\section{Introduction to homotopy theory}
\Frame{Homotopy theory}{
\begin{center}
Study of space or shapes \\
with ``weak equivalences''
\bigskip
\td{plaatje}
\end{center}
}
\Frame{Important spaces}{
\td{plaatjes}
\begin{align*}
S^1 &= ... \\[1em]
S^2 &= ... \\[1em]
S^3 &= \cdots \\[1em]
&\cdots
\end{align*}
}
\Frame{Important tool}{
Fundamental group:
\[ \pi_1(X) = \text{maps } S^1 \to X \text{ up to homotopy} \]
\bigskip
\td{plaatje}
}
\Frame{Important tools}{
Homotopy groups:
\begin{align*}
\pi_1(X) &= \text{maps } S^1 \to X \text{ up to homotopy} \\[1em]
\pi_2(X) &= \text{maps } S^2 \to X \text{ up to homotopy} \\[1em]
\pi_3(X) &= \text{maps } S^3 \to X \text{ up to homotopy} \\[1em]
&\cdots
\end{align*}
}
\Frame{Torsion-free}{
Serre proved in 1950s:
\begin{align*}
\text{odd } k: \quad \pi_n(S^k) \tensor \Q &=
\begin{cases}
\Q &\text{ if } n = k \\
0 &\text{ otherwise }
\end{cases} \\[1em]
\text{even } k: \quad \pi_n(S^k) \tensor \Q &=
\begin{cases}
\Q &\text{ if } n = k, 2k-1 \\
0 &\text{ otherwise }
\end{cases} \\
\end{align*}
}
\section{Rational homotopy theory}
\Frame{Rational homotopy theory}{
\begin{center}
Study of spaces\\
with ``rational equivalences'' \\
and ``rational homotopy groups''
\pause
\bigskip
or
\bigskip
Study of \emph{rational} spaces \\
with weak equivalences \\
and ordinary homotopy groups
\end{center}
}
\Frame{Rational spaces}{
$X$ is \emph{rational} if $\pi_n(X)$ is a $\Q$-vector space
\bigskip
\td{plaatje telescoop}
}
\section{The main equivalence}
\Frame{Main equivalence}{
\begin{theorem}
\begin{center}
Homotopy theory of rational spaces \\
= \\
Homotopy theory of commutative differential graded algebras
\end{center}
\end{theorem}
}
\Frame{Main equivalence (precise version)}{
\begin{theorem}
\[ \Ho(\Top_{\Q, 1, f}) \simeq \opCat{\Ho(\CDGA_{\Q, 1, f})} \]
\end{theorem}
}
\Frame{What is a cdga?}{
\begin{definition}
a cdga $A$ is
\begin{itemize}
\item a $\Q$-vector space
\item with a multiplication $A \tensor A \tot{\mu} A$
\item with a differential $A \tot{d} A$ such that $d^2 = 0$
\item with a grading $A = \bigoplus_{n \in \N} A^n$
\item it is commutative: $ x y = (-1)^{\deg{x}\cdot\deg{y}} y x $
\end{itemize}
\end{definition}
}
\Frame{Free cdga's}{
As always: there is a free guy: $\Lambda(...)$
For example
\[ \Lambda(t, dt) \text{ with } \deg{t} = 0 \]
is just polynomials in $t$, with its differential $dt$
}
\newcommand{\Dict}[1]{
\noindent
\begin{tabularx}{\textwidth}{ X X }
{\bf rational spaces} & {\bf cdga's} \\[1em]
#1
\end{tabularx}
}
\Frame{Dictionary}{
\Dict{
$S^n$ with $n$ odd
& $\Lambda(e)$ with $\deg{e} = n$ \\[1em]
$S^n$ with $n$ even
& $\Lambda(e, f)$ with $\deg{e} = n$, $\deg{f} = 2n-1$ and $d f = e^2$ \\[1em]
Eilenberg-MacLane space $K(\Q, n)$
& $\Lambda(e)$ with $\deg{e} = n$
}
}
\Frame{Dictionary}{
\Dict{
homotopy $$h: X \times I \to Y$$
& homotopy $$h: A \to B \tensor \Lambda(t, dt)$$ \\[1em]
$f: X \to Y$ weak equivalence if $\pi_n(f): \pi_n(X) \iso \pi_n(Y)$
& $f: A \to B$ weak equivalence if $H(f): H(X) \iso H(Y)$
}
}
\end{document}

160
presentation/preamble.tex Normal file
View file

@ -0,0 +1,160 @@
% normally included with amsart
% \usepackage{amsmath, amsthm}
% font with unicode support, does not work with classicthesis
% \usepackage{fontspec}
% clickable tocs
\usepackage{hyperref}
% floating figures
\usepackage{float}
% for multiple cites
\usepackage{cite}
% fancy diagrams
\usepackage{tikz}
\usetikzlibrary{matrix, arrows, decorations}
\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto}
% simple diagrams
% \usepackage[all,cmtip]{xy}
\usepackage{graphicx}
% \graphicspath{ {./images/} }
\usepackage{caption}
\usepackage{subcaption}
% for the fib arrow
\usepackage{amssymb}
% Some basic objects
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\Np}{{\mathbb{N}^{>0}}} % positive numbers
\newcommand{\Z}{\mathbb{Z}} % integers
\newcommand{\R}{\mathbb{R}} % reals
\DeclareRobustCommand{\Q}{\mathbb{Q}} % rationals
\renewcommand{\k}{\mathrm{I\!k}} % default ground ring
% Basic category stuff
\newcommand{\cat}[1]{\mathbf{#1}} % the category of ...
\newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category
\newcommand{\Hom}{\mathbf{Hom}}
\newcommand{\id}{\mathbf{id}}
\newcommand{\Ho}{\cat{Ho}}
% Categories
\newcommand{\Set}{\cat{Set}} % sets
\newcommand{\Top}{\cat{Top}} % topological spaces
\newcommand{\Grp}{\cat{Grp}} % groups
\newcommand{\Ab}{\cat{Ab}} % abelian groups
\newcommand{\DELTA}{\boldsymbol{\Delta}}% the simplicial cat
\newcommand{\simplicial}[1]{\cat{s{#1}}}% simplicial objects
\newcommand{\sSet}{\simplicial{\Set}} % simplicial sets
\newcommand{\Mod}[1]{\cat{{#1}Mod}} % modules over a ring
\newcommand{\Alg}[1]{\cat{{#1}Alg}} % algebras over a ring
\newcommand{\grMod}[1]{\cat{gr\mbox{-}{#1}Mod}} % graded modules over a ring
\newcommand{\grAlg}[1]{\cat{gr\mbox{-}{#1}Alg}} % graded algebras over a ring
\newcommand{\dgMod}[1]{\cat{dg\mbox{-}{#1}Mod}} % differential graded modules over a ring
\newcommand{\dgAlg}[1]{\cat{dg\mbox{-}{#1}Alg}} % differential graded algebras over a ring
\newcommand{\Ch}[1]{\cat{Ch_{n\geq0}({#1})}} % chain complexes
\newcommand{\CoCh}[1]{\cat{Ch^{n\geq0}({#1})}} % cochain complexes
\DeclareRobustCommand{\DGA}{\cat{DGA}} % cochain algebras
\DeclareRobustCommand{\CDGA}{\cat{CDGA}} % commutative cochain algebras
\DeclareRobustCommand{\AugCDGA}{\cat{CDGA^\ast}}% augmentedcommutative cochain algebras
\newcommand{\cof}{\hookrightarrow} % cofibration
\newcommand{\fib}{\twoheadrightarrow} % fibration
\newcommand{\we}{\tot{\simeq}} % weak equivalence
% for use in xy diagrams
\newcommand{\arcof}{\ar@{^{(}->}}
\newcommand{\artcof}{\ar@{^{(}->}|\simeq}
\newcommand{\arfib}{\ar@{->>}}
\newcommand{\artfib}{\ar@{->>}|\simeq}
\newcommand{\arwe}{\ar|-\simeq}
\newcommand{\ariso}{\ar|-\iso}
% adjunction symbol for xymatrices
\newcommand{\xyadj}{\raisebox{0.2\height}{\scalebox{0.5}{$\perp$}}}
% pushout and pullback for xymatrices (makes empty arrow with text)
\newcommand{\xypo}{\ar@{}[dr]|(.75){\scalebox{1.2}{$\ulcorner$}}}
\newcommand{\xypb}{\ar@{}[dr]|(.25){\scalebox{1.2}{$\lrcorner$}}}
%\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}}
\newcommand{\leftadj}{\rightleftarrows}
% Notation and operators
\newcommand{\I}{\,\mid\,} % seperator in set notation
\newcommand{\del}{\partial} % boundary
\newcommand{\iso}{\cong} % isomorphic
\newcommand{\eq}{\sim} % homotopic
\newcommand{\ison}[1]{\stackrel{(#1)}{\iso}} % isos to refer to
\newcommand{\refison}[1]{{\small $(#1)$}} % ref
\newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} % arrow with name
\newcommand{\toti}[1]{\xleftarrow{\,\,{#1}\,\,}} % left arrow with name
\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} % mapsto with name
\newcommand{\unit}{\eta}
\newcommand{\counit}{\epsilon}
\DeclareMathOperator*{\im}{im}
\DeclareMathOperator*{\coker}{coker}
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator*{\Tor}{Tor}
\DeclareMathOperator*{\Ext}{Ext}
\DeclareMathOperator*{\tensor}{\otimes}
\DeclareMathOperator*{\bigtensor}{\bigotimes}
\renewcommand{\deg}[1]{{|{#1}|}}
\newcommand{\Char}[1]{char({#1})}
\newcommand{\RH}{\widetilde{H}} % reduced homology
\DeclareRobustCommand{\C}{\mathcal{C}} % Serre mod C class
\newcommand{\Apl}[0]{{A_{PL}}} % Apl simplicial set of polynomials
\newcommand{\titleCDGA}{\texorpdfstring{$\CDGA$}{CDGA}}
% restriction of a function
\newcommand\restr[2]{{% we make the whole thing an ordinary symbol
\left.\kern-\nulldelimiterspace % automatically resize the bar with \right
#1 % the function
\vphantom{\big|} % pretend it's a little taller at normal size
\right|_{#2} % this is the delimiter
}}
% Todos in the margin
\newcommand{\todo}[1]{
\addcontentsline{tdo}{todo}{\protect{#1}}
$\ast$ \marginpar{\tiny $\ast$ #1}
}
% Big todos in text
\newcommand{\TODO}[1]{
\addcontentsline{tdo}{todo}{\protect{#1}}
{\tiny $\ast$ #1}
}
% TODO item, as itemize does not work
\newcommand{\titem}[0]{\\-\qquad}
% List of todos
\makeatletter
\newcommand \listoftodos{\section*{Todo list} \@starttoc{tdo}}
\newcommand\l@todo[2]{
\par\noindent \textit{#2}, \parbox{10cm}{#1}\par
}
\makeatother
% simple way to center an image
\newcommand{\cimage}[2][]{
\begin{center}
\includegraphics[#1]{#2}
\end{center}
}
% simple way to center a diagram
\newcommand{\cdiagrambase}[1]{
\begin{displaymath}
\input{#1}
\end{displaymath}
}
\newcommand{\cdiagram}[1]{
\cdiagrambase{diagrams/#1}
}