\documentclass{beamer}
% $Header: /cvsroot/latex-beamer/latex-beamer/solutions/generic-talks/generic-ornate-15min-45min.en.tex,v 1.4 2004/10/07 20:53:08 tantau Exp $
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% This file is a solution template for:

% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.



% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.


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%\title % (optional, use only with long paper titles)
%{Fourier transform of algebraic measures}
%
%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
\author % (optional, use only with lots of authors)
{A. Aizenbud}
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%   affiliation.


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{Weizmann Institute of Science}
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\date[] % (optional%{August 14, 2008}

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\title % (optional, use only with long paper titles)
[Decimal representation, Cantor set and p-adic numbers.]
{Decimal representation, Cantor set, p-adic numbers and Ostrowski Theorem.}
%
%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
%\author % (optional, use only with lots of authors)
%{A. Aizenbud}
% - Use the \inst{?} command only if the authors have different
%   affiliation.


%\institute[Massachusetts Institute of Technology] % (optional, but mostly needed)
%{Massachusetts Institute of Technology}

%\myD
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}


\begin{frame}
\myTP
%\center{Joint with: Dmitry Gourevitch}

$ $

$ $

\center{\url{http://www.wisdom.weizmann.ac.il/~aizenr}}

\end{frame}
\begin{frame}
\myFT{Approaches to the real numbers}
%\myP
\begin{itemize}
\myIt axiomatic approach
\myIt Cauchy sequences
\myIt Dedekind sections
\myIt decimal representations
\end{itemize}
\myP
The last is the most elementary but have some problems:
\begin{itemize}
\myIt Hard to perform simple operations: \myP
$$0.666\cdots+0.33\cdots33233\cdots=0.\cdots$$
\myP
\begin{center}
but
\end{center}
$$0.666\cdots+0.33\cdots33433\cdots=1.\cdots$$
\myIt Multiple representation:\myP
$$0.9999\cdots=1.000\cdots$$

\end{itemize}
\end{frame}

\begin{frame}
\myFT{Decimal representation}
\myP
Decimal representation can be viewed as  a map
$$D:\{0,\dots,9\}^\N \to [0,1].$$ \myP
This map is
\begin{itemize}
\myIt Onto
\myIt Continuous
\myIt A topological quotient
\end{itemize}
\myP
\begin{center}
\color{red}{BUT NOT 1-1}
\end{center}
\myP
{\color{blue}
If it would be 1-1 it would be an homeomorphism.}
\end{frame}

\begin{frame}
\myFT{The Cantor set}
\myP
\begin{exercise}
$\{0,\dots,9\}^\N \cong \myP \{0,\dots,n\}^\N \cong\myP \prod F_i \cong \myP \{(0.a_1\cdots)_3|a_i\in\{0,1\}\} \myP \cong   \text{ Cantor set }$
\end{exercise}
%\includegraphics{729px-Cantor_set_in_seven_iterations.svg}
\myP
\includegraphics[width=10cm]{Cantor_set_constr.png}
\end{frame}

\begin{frame}
\myFT{p-adic numbers}
\myP
\begin{definition}
$$\Q_p=\{(\cdots a_0.a_1\cdots a_n )|a_{i}\in\{0,\dots,p-1\}\}$$
\begin{itemize}
\myIt $+,-,\cdot$ are  easy to define
\end{itemize}
\end{definition}
\myP
\begin{exercise}
One can define ``$:$'' iff $p$ is prime.
\end{exercise}
\myP
\begin{definition}[p-adic absolute value]
$|p^n \cdot \frac{a}{b}|_{p}:=p^{-n}$, where $(a,p)=(b,p)=1$
\end{definition}
\myP
\begin{exercise}
Equivalent definition of $\Q_p$: Completion of $\Q$ w.r.t. $|\cdot|_p$
\end{exercise}
\end{frame}
\begin{frame}
\myFT{Absolute Value}
\myP
\begin{definition}[absolute value]
\begin{itemize}
\item absolute value on a field $F$ is a map $|\cdot|: F \to \R_{\geq 0}$ s.t.
\begin{itemize}
\myIt $|a|=0$ iff a=0,
\myIt $|ab|=|a||b|$,
\myIt $|a+b|\leq |a|+|b|$.
\end{itemize}
\myIt $|\cdot|_\alpha$ and
 $|\cdot|_\beta$ are equivalent iff they define the same topology (i.e. the same notion of convergence)
\end{itemize}
\end{definition}
\myP
\begin{exercise}
\begin{itemize}
\myIt  $|\cdot|_\alpha$ and
 $|\cdot|_\beta$ are equivalent iff $|\cdot|_\alpha^a=|\cdot|_\beta$
\myIt  We have the following dichotomy
\begin{itemize}
\myIt Either $|\cdot|$ is Archimedean, i.e. $|\Z|$ is unbounded
\myIt Or $|\cdot|$ is non-Archimedean, i.e. $|a+b|\leq \max(|a|,|b|)$.
\end{itemize}
\myIt Show that if $|\cdot|$  is non-Archimedean then $|\cdot|^a$ is also a (non-Archimedean) absolute value
\end{itemize}
\end{exercise}
\end{frame}


\begin{frame}
\myFT{Ostrowski Theorem}
\myP
\begin{theorem}[Ostrowski]
the possible (up to equivalence) absolute values on $\Q$ are:
\begin{itemize}
\myIt Archimedean: the usual one -- $|\cdot|_\infty$.
\myIt non-Archimedean: the p-adic ones -- $|\cdot|_p$.
\end{itemize}
\end{theorem}
\myP
\begin{exercise}
What powers of $|\cdot|_\infty$ are  absolute value
\end{exercise}
\end{frame}


\begin{frame}
\myFT{Local Fields}
\myP
\begin{definition}[Local Field]
A local field is a locally compact non-discrete topological field.
\end{definition}
\myP
\begin{theorem}[Classification of Local Fields]
The possible local fields are:
\begin{itemize}
\myIt positive Characteristic: $\F_q((t))$
\myIt zero Characteristic:
\begin{itemize}
\myIt non-Archimedean: finite extortions of $\Q_p$
\myIt Archimedean:
\begin{itemize}
\myIt $\R$
\myIt $\C$
\end{itemize}
\end{itemize}
\end{itemize}
\end{theorem}
\end{frame}
\end{document}
%\myED


