From 94e6bac00ac674b8722e3b80ece83a94a94d0f45 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Thu, 13 Jan 2022 12:46:06 +0100 Subject: =?UTF-8?q?Wst=C4=99pny=20szkielet=20pracy.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- lic_malinka.tex | 208 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 208 insertions(+) create mode 100644 lic_malinka.tex (limited to 'lic_malinka.tex') diff --git a/lic_malinka.tex b/lic_malinka.tex new file mode 100644 index 0000000..7abd37d --- /dev/null +++ b/lic_malinka.tex @@ -0,0 +1,208 @@ +\documentclass[11pt, a4paper, final]{amsart} +\setlength{\emergencystretch}{2em} + + +\usepackage[T1]{fontenc} +\usepackage{mathtools} +\usepackage[activate={true,nocompatibility},final,tracking=true,kerning=true,spacing=true,stretch=10,shrink=10]{microtype} +\microtypecontext{spacing=nonfrench} +\usepackage[utf8]{inputenc} + +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage{XCharter} +\usepackage[charter,expert, greekuppercase=italicized, greekfamily=didot]{mathdesign} +\usepackage{mathtools} +\usepackage{enumitem} +\usepackage[utf8]{inputenc} +\usepackage{tikz-cd} +\usepackage{tikz} + +\usepackage{etoolbox} + + +\usepackage{xcolor} +\definecolor{green}{RGB}{0,127,0} +\definecolor{redd}{RGB}{191,0,0} +\definecolor{red}{RGB}{105,89,205} +\usepackage[colorlinks=true]{hyperref} + +\usepackage[notref, notcite]{showkeys} +\usepackage[cmtip,arrow]{xy} +%\usepackage[backend=biber, +%url=false, +%isbn=false, +%backref=true, +%citestyle=alphabetic, +%bibstyle=alphabetic, +%autocite=inline, +%maxnames=99, +%minalphanames=4, +%maxalphanames=4, +%sorting=nyt,]{biblatex} +%\addbibresource{linear_strucures.bib} + +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Stab}{Stab} +\DeclareMathOperator{\st}{st} +\DeclareMathOperator{\Flim}{FLim} + +\newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}} +\newcommand{\cC}{\mathcal C} +\newcommand{\bN}{\mathbf N} +\newcommand{\bR}{\mathbf R} +\newcommand{\bZ}{\mathbf Z} + +\DeclareMathOperator{\im}{{Im}} +\DeclareMathOperator{\lin}{{Lin}} +\DeclareMathOperator{\Th}{{Th}} + + +\newtheorem{theorem}{Theorem} +\numberwithin{theorem}{section} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{claim}[theorem]{Claim} +\newtheorem{fact}[theorem]{Fact} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{conjecture}[theorem]{Conjecture} +\newtheorem{axiom}[theorem]{Axiom} +\newtheorem{question}[theorem]{Question} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem*{theorem2}{Theorem} +\newtheorem*{claim2}{Claim} +\newtheorem*{corollary2}{Corollary} +\newtheorem*{question2}{Question} +\newtheorem*{conjecture2}{Conjecture} + + +\newtheorem{clm}{Claim} +\newtheorem*{clm*}{Claim} + + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem*{definition2}{Definition} +\newtheorem{example}[theorem]{Example} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem*{remark2}{Remark} + + +\AtEndEnvironment{proof}{\setcounter{clm}{0}} +\newenvironment{clmproof}[1][\proofname]{\proof[#1]\renewcommand{\qedsymbol}{$\square$(claim)}}{\endproof} + +\newcommand{\xqed}[1]{% + \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill + \quad\hbox{\ensuremath{#1}}} + + +\title{Tytuł} +\author{Franciszek Malinka} + +\begin{document} + + \begin{abstract} + Abstract + \end{abstract} + \section{Introduction} + + \section{Preliminaries} + \subsection{Descriptive set theory} + \begin{definition} + Suppose $X$ is a topological space and $A\subseteq X$. We say that $A$ is \emph{meagre} in $X$ if... Wea say that $A$ is \emph{comeagre} in $X$ if... . + \end{definition} + + \begin{definition} + We say that a topological space $X$ is a \emph{Baire space} if every comeagre subset of $X$ is dense in $X$ (equivalently, every meagre set has empty interior). + \end{definition} + + \begin{definition} + Suppose $X$ is a Baire space. We say that a property $P$ holds generically for a point in $x\in X$ if $\{x\in X\mid P\textrm{ holds for }x\}$ is comeagre in $X$. + \end{definition} + + \begin{example} + content + \end{example} + + \subsection{Fraïssé classes} + \begin{fact}[Fraïssé theorem] + \label{fact:fraisse_thm} + Suppose $\cC$ is a class of finitely generated $L$-structures such that... + + Then there exists a unique up to isomorphism counable $L$-structure $M$ such that... + \end{fact} + + + \begin{definition} + For $\cC$, $M$ as in Fact~\ref{fact:fraisse_thm}, we write $\Flim(\cC)\coloneqq M$. + \end{definition} + + \begin{fact} + If $\cC$ is a uniformly locally finite Fraïssé class, then $\Flim(\cC)$ is $\aleph_0$-categorical and has quantifier elimination. + \end{fact} + + \section{Conjugacy classes in automorphism groups} + + \subsection{Prototype: pure set} + In this section, $M=(M,=)$ is an infinite countable set (with no structure beyond equality). + \begin{proposition} + If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are conjugate if and only if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$ and $f_2$ have the same number of orbits of size $n$. + \end{proposition} + + \begin{proposition} + The conjugacy class of $f\in \Aut(M)$ is dense if and only if... + \end{proposition} + \begin{proposition} + If $f\in \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is meagre. + \end{proposition} + + \begin{proposition} + An automorphism $f$ of $M$ is generic if and only if... + \end{proposition} + + \begin{proof} + + \end{proof} + + \subsection{More general structures} + + + \begin{proposition} + Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong (M,f_2)$. + \end{proposition} + + \begin{definition} + We say that a Fraïssé class $\cC$ has \emph{weak Hrushovski property} (\emph{WHP}) if for every $A\in \cC$ and partial automorphism $p\colon A\to A$, there is some $B\in \cC$ such that $p$ can be extended to an automorphism of $B$, i.e.\ there is an embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following diagram commutes: + \begin{center} + \begin{tikzcd} + B\ar[r,"\bar p"]&B\\ + A\ar[u,"i"]\ar[r,"p"]&A\ar[u,"i"] + \end{tikzcd} + \end{center} + \end{definition} + + \begin{proposition} + Suppose $\cC$ is a Fraïssé class in a relational language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$, all orbits of $f$ are finite. + \end{proposition} + \begin{proposition} + Suppose $\cC$ is a Fraïssé class in an arbitrary countable language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$ ... + \end{proposition} + + \subsection{Random graph} + \begin{definition} + The \emph{random graph} is... + \end{definition} + + \begin{fact} + The + \end{fact} + + \begin{proposition} + Generically, the set of fixed points of $f\in \Aut(M)$ is isomorphic to $M$ (as a graph). + \end{proposition} + +\end{document} -- cgit v1.2.3