%
\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 $
%\texhash

% 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.


\mode<presentation>
{
  \usetheme{Warsaw}
  % or ...
  % or whatever (possibly just delete it)
  \setbeamercovered{invisible}
}

%\usetheme{Boadilla}


\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever
\usepackage[all]{xy}
\usepackage{xypic}
\usepackage{times}
\usepackage{ulem}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
%\newtheorem{theorem*}{Theorem}
%\newtheorem*{lemma}{Lemma}
\newtheorem*{proposition}{Proposition}
\newtheorem*{remark}{Remark}
\newtheorem*{question}{Question}
\newtheorem*{Task}{Task}

%\newtheorem*{example}{Example}
%\newtheorem*{theorem}{Theorem}
%\newtheorem*{definition}{Definition}
\newtheorem*{notation}{Notation}
%\newtheorem*{result}{Result}
%\newtheorem*{property}{}
%\newtheorem*{corollary}{Corollary}
%\newtheorem*{construction}{Construction}
%\newtheorem*{case}{Case}
\newtheorem*{conjecture}{Conjecture}
\newtheorem*{setting}{Setting}
\newtheorem*{flowchart}{Flowchart}

\newcommand{\R}{{\mathbb R}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\C}{{\mathbb C}}
%\newcommand{\Dist}{{\mathcal{D}}}
\newcommand{\Dist}{\Sc^*}

\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Sc}{{\mathcal S}}
\newcommand{\Fre}{{Fr\'{e}chet \,}}
\newcommand{\Fou}{\mathcal F}


\newcommand{\Matrix}[4]{ \left( \begin{array}{cc}  #1 & #2 \\  #3 & #4 \\ \end{array} \right) }
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\Cat}[1]{ {\mathfrak #1} }
\newcommand{\Lie}[1]{ {\mathfrak #1} }
\newcommand{\filt}[1]{ {\mathcal #1} }
\newcommand{\scheme}[1]{\underline{ \mathbf{#1}}}
\newcommand{\alg}[1]{\mathbf{#1}}
\newcommand{\mc}[1]{\mathcal #1}
\newcommand{\Hom}{ {\mbox{Hom}} }
\newcommand{\Ext}{ {\mbox{Ext}} }
\newcommand{\Ind}{ {\mbox{Ind}} }
% \newcommand{\proof}{ {\sc Proof:} }
% \newcommand{\qed}{\begin{flushright}Q.E.D \end{flushright}}
\newcommand{\nats}{\mathbb N}
\newcommand{\ints}{\mathbb Z}
\newcommand{\rats}{\mathbb Q}
\newcommand{\reals}{\mathbb R}
\newcommand{\complex}{\mathbb C}
\newcommand{\quats}{\mathbb H}
\newcommand{\octs}{\mathbb O}
\newcommand{\hecke}{\mathcal H}
\newcommand{\fourier}{\mathcal F}
\newcommand{\FF}{\mathbb F}
\newcommand{\Gp}{\mathbb G}
\newcommand{\adeles}{\mathbb A}
%\newcommand{\eps}{\varepsilon}
\newcommand{\To}{\longrightarrow}
\newcommand{\BX}{\mathbf{B}(X)}

\newcommand{\A}{\mathcal{A}}
\newcommand{\D}{\mathcal{D}}
\renewcommand{\cH}{\mathcal{H}}


\newcommand{\DOP}{\mathbb{D}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\Schwarz}{\mathcal{S}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\OO}{\mathcal{O}}
\newcommand{\ZU}{\mathcal{Z}}
\newcommand{\e}{\textbf{e}}
\newcommand{\Orb}{\mathcal{O}}
\newcommand{\Period}{\mathcal{P}}
\newcommand{\isom}{\cong}
\newcommand{\gK}{( \Lie{g}_\complex, K)}
\newcommand{\J}{\mathcal J}
\newcommand{\g}{{\mathfrak{g}}}
\newcommand{\N}{{\mathcal N}}
\newcommand{\h}{{\mathfrak{h}}}




\newcommand{\speh}{\operatorname{Speh}}
\newcommand{\aaa}{\mathfrak{a}}
\newcommand{\res}{\operatorname{res}}
\newcommand{\mix}{\mathcal{M}}
\newcommand{\tr}{\operatorname{Tr}}
\newcommand{\www}{\mathfrak{W}}
\newcommand{\ob}{\operatorname{ ob}}
\newcommand{\bes}{\mathfrak{B}}
\newcommand{\eisen}{\mathcal{E}}
\newcommand{\rest}{\lvert}
\newcommand{\sg}{\mathfrak{S}}
\newcommand{\proj}{\operatorname{proj}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\disc}{\Delta}
\newcommand{\modulus}{\delta}
\newcommand{\I}{\mathbb{I}}
\newcommand{\eps}{\varepsilon}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\Ht}{H}
\newcommand{\diff}{\mathfrak{d}}
\newcommand{\triv}{{\bf 1}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\inv}{\operatorname{inv}}
%\newcommand{\Ind}{\operatorname{Ind}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\match}[1]{\mathop{\leftrightarrow}\limits^{#1}}
\newcommand{\der}[1]{\mathfrak{D}^{(#1)}}
%\newcommand{\abs}[1]{\left|{#1}\right|}
%\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\sprod}[2]{\left\langle#1,#2\right\rangle}


\newcommand{\ff}{\mathbb{F}}
\newcommand{\bbc}{\mathbb{C}}
\newcommand{\bc}{\operatorname{bc}}
\newcommand{\ad}{\operatorname{ad}}
\newcommand{\uu}{\mathcal{U}}
%\newcommand{\hecke}{\mathcal{H}}
\newcommand{\pp}{\mathcal{P}}
\newcommand{\oo}{\mathcal{O}}
\newcommand{\ch}{\operatorname{ch}}
\newcommand{\sss}{\mathcal{S}}
\newcommand{\ww}{\mathcal{W}}
\newcommand{\bb}{\mathcal{B}}
\newcommand{\ii}{\mathcal{I}}
\newcommand{\ccc}{\mathcal{C}}
\newcommand{\nm}{\operatorname{Nm}}
\newcommand{\bs}{\backslash}
%\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\kk}{\mathfrak{K}}
\newcommand{\zz}{\mathbb{Z}}
\newcommand{\nn}{\mathbb{N}}
\newcommand{\cc}{\mathbb{C}}
\newcommand{\qq}{\mathbb{Q}}
\newcommand{\rr}{\mathbb{R}}
\newcommand{\aut}{\mathcal{A}}
\newcommand{\Supp}{\mathrm{Supp}}
%\newcommand{\g}{{\mathfrak{g}}}
\newcommand{\proofend}{\hfill$\Box$\smallskip}
\definecolor{DarkGreen}{rgb}{0,0.5,0}
\newcommand{\Rami}[1]{{\color{red}{#1}}}

%\newcommand{\A}{\mathbb{A}}
\title % (optional, use only with long paper titles)
{Cohen macauly property in representation theory}
%
%%\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}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[] % (optional)
%{August 14, 2008}

%\subject{}
% This is only inserted into the PDF information catalog. Can be left
% out.



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:
%
%\pgfdeclareimage[height=0.5cm]{fiatslug}{fiatslug}
%\logo{\pgfuseimage{fiatslug}}



% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
%\AtBeginSubsection[]
%{
%  \begin{frame}<beamer>
%    \frametitle{Outline}
%    \tableofcontents[currentsection,currentsubsection]
%  \end{frame}
%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:

%\beamerdefaultoverlayspecification{<+->}


\begin{document}

\begin{frame}
\titlepage
\center{Joint with E. Sayag} \\$ $ \\ $ $ \\
\center{\url{http://math.mit.edu/~aizenr} }

\end{frame}
%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
%  % You might wish to add the option [pausesections]
%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Part I: Overview}
%
%\subsection{Gelfand Pairs}
\begin{frame}
    \frametitle{Some Phenomena in Harmonic Analysis}
\thispagestyle{empty}
\begin{small}
\pause
Let $G \supset H$ be a (symmetric) pair or reductive groups over a non-Archimedean local field.
\pause
\begin{example}[Behavior of multiplicity]
\pause
Let
\begin{itemize}
 \item 
 $\pi_\nu$ be a family of admissible representations, 
\pause
e.g. $\pi_\nu=Ind_P^G(\rho \otimes \chi_\nu)$.
\pause
 \item $m_\nu:=\dim \Hom_H(\pi_\nu,\C)$
\pause -- this is an upper-semi-continuous function.
\end{itemize}
\pause
Sometimes it is continuous.
\end{example}
\pause
\begin{example}[Density of orbital regular integrals]
Often:
$\overline{\sum_{g \in G^{H\times H - reg}}  \Sc^*(HgH)^{H\times H}}=\Sc^*(G)^{H\times H}$
\end{example}
\pause
\begin{example}[Freeness of the Hecke module]
Let $K<G$ be a compact open subgroup. 
\pause
Sometimes $\Sc(G/H)^K$ is free over the center of the Hecke algebra $\cH(G,K)$.
\end{example}
\end{small}
\end{frame}


\begin{frame}
    \frametitle{Cohen Macaulay Property}
%\thispagestyle{empty}
%\begin{small}
\pause
\begin{theorem}
Let $A$  be a finitely generated commutative algebra.\pause
Let  $M$ be a finitely generated module over it. \pause
Then the following are equivalent:
\pause
\begin{itemize}
 \item $M$ is f.g. and (locally) free over some polynomial subalgebra of $A$.
 \pause
 \item If $B \subset A$ is a polynomial subalgebra and $M$ is f.g. over $B$, then $M$ (locally) free over $B$.
\pause
 \item $M$ is f.g. and locally free over some regular subalgebra of $A$.
 \pause
 \item If $B \subset A$ is a regular subalgebra and $M$ is f.g. over $B$, then $M$ is locally free over $B$.
 \end{itemize}
\end{theorem}
\pause
 \begin{definition}
In this case $M$ is called a Cohen-Macaulay module over $A$.
\end{definition}

%\end{small}
\end{frame}


\begin{frame}
    \frametitle{The Main Conjecture}
%\thispagestyle{empty}
%\begin{small}
\pause
\begin{conjecture}
 Let $X$ be a (symmetric) $G$-space. Then

$\Sc(X)$ is a Cohen-Macaulay object in the category of smooth $G$-modules.
\end{conjecture}
\end{frame}
\end{document}
