\documentclass{beamer}

% 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}
}

%\usepackage[english]{babel}
%\usepackage{times}
\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever

    \usepackage[all]{xy}
    \usepackage{xypic}
    \usepackage{times}
    \usepackage{ulem}
    \usepackage[T1]{fontenc}
    \usepackage{amsfonts,amsmath,amssymb}
    \usepackage{hyperref}
    %\usepackage[all]{xy}
    \usepackage{amssymb,amsthm,amsxtra}
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    \usepackage{mathdots}


\newtheorem*{proposition}{Proposition}
\newtheorem*{prop}{Proposition}
\newtheorem*{remark}{Remark}
\newtheorem*{rem}{Remark}
\newtheorem*{question}{Question}
\newtheorem*{mypicture1}{Situation 1}
\newtheorem*{mypicture2}{Situation 2}
\newtheorem*{keylemma}{Key Lemma}
\newtheorem*{cokeylemma}{co-Key Lemma}

%\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}

%\newtheorem{cor}[lemma]{Corollary}
%\newtheorem{exm}[lemma]{Example}
%\newtheorem{exc}[lemma]{Exercise}
%\newtheorem{conj}[lemma]{Conjecture}
%\newtheorem{rem}[lemma]{Remark}
%\newtheorem{conc}[lemma]{Conclusion}



\newcommand{\F}{{\mathbb F}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Dist}{{\mathcal{D}}}
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\newcommand{\Sc}{{\mathcal S}}
\newcommand{\Fre}{{Fr\'{e}chet \,}}


\newcommand{\Matrix}[4]{ \left( \begin{array}{cc}  #1 & #2 \\  #3 & #4 \\ \end{array} \right) }
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
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\newcommand{\Hom}{ {\mbox{Hom}} }
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% \newcommand{\proof}{ {\sc Proof:} }
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\newcommand{\Rami}[1]{{\color{cyan}{#1}}}
\newcommand{\Dima}[1]{{\color{magenta}{#1}}}
\newtheorem{observation}{Observation}
\newtheorem{conclution}{Conclusion}
\newtheorem{conclusion}{Conclusion}


% You can add any graphics to every slide by following command:
% \logo{\includegraphics{logo.eps}}

% Uncomment this, if you want the table of contents before each subsection.
% However, to edit slides in TeXWord avoiding this feature is good idea.
% \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}

% Title Data. We keep it after \begin{document}
% to enable editing text in BaKoMa TeX Word.

\title{Smooth Transfer of Kloosterman Integrals}

% Use the \inst command to identify several affiliations.
\author{Avraham Aizenbud}
\institute{Department of Mathematics, Massachusetts Institute of Technology}
%Dima? I think ``Department of Mathematics'' is irrelevant.
%But if you want it to stay - separate it by \\ or by ,


%\date[TUG 2005]{Annual TUG Conference, 2005}

%\subject{Desktop Publishing} % Should be passed to PDF [YNI]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
\begin{frame}
%\maketitle
\titlepage
\begin{center}Joint with Dmitry Gourevitch \end{center}
$ $\\
\begin{center}\url{http://math.mit.edu/~aizenr/}  \end{center}

\end{frame}
%\newpage
\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Motivation}
\pause
\center{Representation Theory of $G$}
\pause
$$\Updownarrow$$
\center{Harmonic analysis on $G$ w.r.t. the two-sided action of $G \times G$}
\pause
$$\Uparrow$$
\center{Harmonic analysis on $G/H$ w.r.t. the action of $G$}
\pause
$$\Updownarrow$$
\center{Study of $End(\Sc(G/H))$}
\pause
$$\Uparrow$$
\center{Study of $Hom(\Sc(G/H_1),\Sc(G/H_2))$}
\pause
$$\Updownarrow$$
\center{Study of $\Sc^*((G/H_1) \times (G/H_2))^G$}
\pause
$$\Uparrow$$

\end{frame}
\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Motivation}
$$\Uparrow$$
\center{Study of $\Sc^*(X)^G$}
\pause
$$\Updownarrow$$
\center{Study of $\Sc(X)_G$}
\pause
$$\text{ \quad \quad \quad \quad \quad \quad \quad }\Updownarrow \text{ (in many cases)}$$
$\text{\quad \quad \quad \quad \quad}$ Study of the image of $\Omega_G: \Sc(X) \to F(X//G)$

\end{frame}
\begin{frame}
    \frametitle{Real Life}
\pause
%\center{
We want to compare representation theory of $G$ and $G'$.
%}
\pause

$ $\\
%$ $\\ $ $ \\
%\noindent
Step 1: Smooth transfer -- \\ Comparison of  $\Omega_{Ad(G)}: \Sc(G) \to F(G//Ad(G))$ and $\Omega_{Ad(G')}: \Sc(G') \to F(G'//Ad(G')).$
\pause

$ $\\
Step 2: Comparison of representation theories --
%\center{Using smooth transfer: comparison of $\Omega_{Ad(G)}(\Sc(G))$ and $\Omega_{Ad(G')}(\Sc(G'))$}
%\pause
%\center{
%Very difficult but possible
%}
\pause
%\center{
Requires global (adelic) arguments and the trace formula.
%}

\end{frame}

\begin{frame}
    \frametitle{Real Life}
We want to compare harmonic analysis on $G$-spaces and $G'$-spaces.

$ $\\
Step 1: Smooth transfer -- \\ Comparison of  $\Omega_{H_1 \times H_2}: \Sc(G) \to F(H_1 \setminus \setminus G//H_2)$ and $\Omega_{H'_1 \times H'_2}: \Sc(G) \to F(H'_1 \setminus \setminus G'//H'_2)$

$ $\\
Step 2: Comparison of harmonic analysis  --
Requires global (adelic) arguments and the relative trace formula.

\pause

\center {We need adelic smooth transfer}
\pause
$$\Updownarrow$$
\center{$\text{ \quad \quad \quad \quad }$  Local smooth transfer $\&$ Fundamental Lemma}

\end{frame}

\begin{frame}
    \frametitle{History}
\center {{\bf Group Case}}
\pause
\begin{itemize}
\item  1970's $GL_2$ vs. $D$ -- Jacquet-Langlands
\pause
\item  1980's $GL_n(F)$ vs. $GL_n(E)$ -- Arthur-Clozel
\pause
\item  1990-2010 Classical groups -- \pause
 Trace formula: Arthur,...;
 \pause
Smooth transfer $\&$ Fundamental Lemma: Waldspurger, Ngo,...
\end{itemize}
\pause
\center {{\bf Relative Case}}
\begin{itemize}
\pause
\item  1990' Rank 1 groups -- Jacquet
\pause
\item  2000' \pause $N(F)\!\setminus \! GL_n(F)/N(F)$ vs. $N(E) \!\setminus \! GL_n(E)/U$ \pause -- Trace formula: \pause Geometric side -- Jacquet, \pause Spectral
 side -- Lapid;
% Kuznetsov
\pause
Fundamental Lemma: Ngo, Jacquet;
\pause Smooth Maching:
\pause p-adic case -- Jacquet,
\pause Real case -- Aizenbud-Gourevitch
\end{itemize}


\end{frame}


\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Introduction}
Fix a natural number $n$, a local field $F$ and its quadratic extension $E$.
\pause
\begin{mypicture1}
$U(F) \times U(F)$ acts on $GL_n(F)$, where
\begin{itemize}
\item  $U$ is the group of upper unipotent matrices of size n.
\item  The action is given by $(u_1,u_2)(g)=u_1^t g u_2$
\end{itemize}
\end{mypicture1}
\pause
\begin{mypicture2}
$U(E)$ acts on $H$, where
\begin{itemize}
\item  $H$ is the space of non-degenerate hermitian matrices of size n.
\item  The action is given by $u (g)=\bar{u}^t g u$
\end{itemize}
\end{mypicture2}
%\begin{theorem}
%$$\widetilde{\Omega}(\Sc(GL_n(F)))=\widetilde{\Omega}(\Sc(H))$$
%\end{theorem}

\end{frame}
\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Notation}

Fix a natural number $n$, a local field $F$ and its quadratic extension $E$.
\pause
\begin{itemize}
\item $M$ is the space of matrices of size $n$ over $F$.
\pause
%\item Denote by $\Delta_i:M \to F$ the main $i$-minor.
%\pause
\item $N:= U(F) \times U(F)$
\pause
\item Fix an additive character $\psi$
\pause
\item We define a character $\theta : N \to
\C^{\times}$ by $\theta(x,y) := \psi(\sum_{i=1}^{n-1} (x_{i,i+1} +
{ y_{i,i+1}}))$.
\pause
\item Denote by $T < GL_n(F)$ the torus.
\pause
\item Define $\Omega: \Sc(M) \to C^{\infty}(T)$
by
$$ \Omega(\Phi)(a):= \int_{N}  \Phi(g a) \theta(g)dg.$$
\pause
%\item Define $\sigma:M \to F$ by $\sigma (x) :=
%\prod_{i=1}^{n-1} \Delta^n_i(x)$.
%\pause
\item
$ \widetilde{\Omega}(\Psi)(a):= \sigma(a)\Omega(\Psi)(a)$
\end{itemize}
\pause
\begin{theorem}
$$\widetilde{\Omega}(\Sc(M))=\widetilde{\Omega}^E(\Sc(M^E))$$
\end{theorem}

\end{frame}

%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Main ingredients}
\pause

\begin{itemize}
\item Induction
\pause
\item The implication of the $GL_n$ case
\pause
\item Partial Kloosterman integrals.
\pause
\item Fourier  transform, Jacquet transform and the inversion formula
\pause
\item Key lemma
\end{itemize}
\end{frame}
%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Partial Kloosterman integrals}
\begin{itemize}
\item Denote by $\Delta_i:M \to F$ the main $i$-minor.
\pause

\item $O_i=\{x\in M| \Delta_i \neq 0\} \subset M$
\pause
\item $$U_i :=
\left \{ \begin{pmatrix}
  Id_{i} & * \\
  0 & Id_{n-i}
\end{pmatrix}
\right \}
< U$$
\item $N_i=U_i(F)\times U_i(F)$
\pause
\item Define $\Omega_i: \Sc(O_i) \twoheadrightarrow \Sc (GL_i \times
M_{n-i})$, by
$$ \Omega_{i}(\Psi)(a):= \int_{N_i} \Psi(u a) \theta(u)du$$
\pause
\end{itemize}

\begin{proposition}
$$\xymatrix{
\parbox{25pt}{$\Sc(O_i)$}\ar@/^2pc/@{->}^{{\Omega}}[rr]\ar@{->}^{\Omega_i \quad \quad \quad}[r] &
\parbox{81pt}{$\Sc(GL_i \times M_{n-i})$} \ar@{->}^{\quad \quad \quad {\Omega}}[r] &
\parbox{25pt}{$C^{\infty}(T)$}}$$
\end{proposition}

\end{frame}
%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Fourier transform, Jacquet transform and the inversion formula}

Fix $B$, a quadratic $F$-form on $M$ given by $B(x,y):= \tr_{F}(x w yw)$
\pause
\begin{theorem}[Jacquet]
$$\xymatrix{\parbox{25pt}{$\Sc(M)$}\ar@{->}^{\mathcal{F} }[r]\ar@{->}_{\widetilde{\Omega}}[d] &
\parbox{25pt}{$\Sc(M)$}\ar@{->}^{\widetilde{\Omega}}[d]\\
%
\parbox{30pt}{$C^{\infty}(T)$}\ar@{->}^{\mathfrak{J}}[r] &
\parbox{30pt}{$C^{\infty}(T)$}}$$
\end{theorem}
\end{frame}
%\newpage

\begin{frame}
\thispagestyle{empty}
%
    \frametitle{Key lemma}
\begin{itemize}
\item Let $V:= \cup O_i$.
\pause

\item Let $Z:=M -  V$.
\pause
%\item
\end{itemize}
\begin{keylemma}
$$\Sc(M)=\Sc(V)+\mathcal{F}(\Sc(V))+\Sc(M)_{(N,\theta)}$$
\end{keylemma}
\pause
\begin{cokeylemma}
$$\Sc^*_M(Z)^{N,\theta} \cap \mathcal{F}(\Sc^*_M(Z)^{N,\theta})=0$$
\end{cokeylemma}
\pause
% sadadsas
%{\small sadadsas}
%{\footnotesize  sadadsas}
\begin{proof}
\begin{itemize}
{\footnotesize
\item
$
Z':=\left\{
\begin{pmatrix}
  0 &  \cdots & 0&a \\
  \vdots& \iddots & \iddots & * \\
  0 &a & \iddots & \vdots \\
  a & * & \cdots & *
\end{pmatrix}
\right\} \subset Z
$
\pause
\item
$\Sc^*_M(Z)^{N,\theta} \subset \Sc^*_M(Z')$
\pause
\item $\Sc^*_M(Z') \cap \mathcal{F}(\Sc^*_M(Z'))=0$
}
\end{itemize}
\end{proof}

\end{frame}
%\newpage

\begin{frame}
%%\thispagestyle{empty}
%
    \frametitle{Difficulties in the Archimedean case}
\begin{enumerate}
\item Topology on $\Sc(M)$
\pause
$\Rightarrow$ we need infinite procedures.
\pause
e.g.:
\begin{itemize}
\item The implication of the $GL$ case,
\pause
\item $\Sc(M) \otimes \Sc(GL_n) \neq \Sc(M \times GL_n)$
%= \Sc(M \widehat{\otimes} GL_n)

\end{itemize}
\pause
\item We do not have Bernstein's localization principle
\pause
\item Transversal derivatives
\pause
\item Key Lemma $\nLeftrightarrow$ co-Key Lemma
\end{enumerate}

\end{frame}
%\newpage

\begin{frame}
\thispagestyle{empty}
%
    \frametitle{Methods to overcome the difficulties}
\begin{enumerate}

\item Theory of Schwartz  functions,  Nuclear Frechet spaces, Schwartz Kernel theorem, Banach open map theorem, Dixmier-Malliavin Theorem.
\pause
\begin{itemize}
\item $0\to \Sc^*_X(Z) \to \Sc^*(X) \to \Sc^*(U) \to 0$
\pause
\item $gr_k(\Sc^*_X(Z))= \Sc^*(Z, Sym^k(CN_{Z}^X))$
\pause
\item $\Sc(X\times Y)=\Sc(X)\hat{\otimes} \Sc(Y)$
\pause
\item $\Sc(\bigcup U_i)=\sum \Sc(U_i)$
\pause

\item For $\phi: X\to Y$ we have $\Sc(X)=\Sc(X)\phi^* (\Sc(Y))$
\end{itemize}
\pause
\item $N$ is unipotent $\Rightarrow$ no non-trivial irreducible algebraic representations of $N$.
\pause
\item Tools dual to distributional tools:
\end{enumerate}
\begin{lemma}[E. Borel]
Let $Z \subset X$ closed subvariety $\Sc(X)/\Sc(X-Z)=\lim \limits_\leftarrow (\Sc(X)/\Sc(X-Z))_{n}$
\end{lemma}
\pause
\begin{example}
$\Sc(\R)/\Sc(\R^\times)=\C[[t]]$
\end{example}


%%%%%%%%%%
\end{frame}
%\newpage

\begin{frame}
%\thispagestyle{empty}
%%%%%%%%%%

%    \frametitle{methods to overcome the dificultis}

\begin{lemma}
Let $W \subset V$ linear spaces s.t. $W \nsupseteq W^{\bot}$ . Then $$\Sc(V)= \Sc(V - W)+\mathcal{F}(\Sc(V - W))$$
\end{lemma}
\pause
\begin{example}
$$\Sc(\R)=\Sc(\R^\times)+\mathcal{F}(\Sc(\R^\times))$$
\end{example}
\pause
We need dual
of the following lemma
\begin{theorem}
Let $G$ act on $X,E$ and $Z \subset X$ closed subvariety  s.t. $$\forall z \in Z,k \in \Z_{\geq 0} :
(E|_z^* \otimes Sym^k(N_{z,Gz}^X) \otimes \Delta_{G,G_z})^{G_z} =0.$$
Then $$\Sc^{*}_{X}(Z,E)^{G}=0.$$
\end{theorem}
%%%%%%%%%%
\end{frame}
%\newpage

\begin{frame}
%\thispagestyle{empty}
%%%%%%%%%%


\begin{theorem}
Let $G$ act on $X,E$ and $Z \subset X$ closed subvariety  s.t. $$\forall z \in Z,k \in \Z_{\geq 0} :
(E|_z \otimes Sym^k(CN_{z,Gz}^X) \otimes \Delta_{G,G_z})_{\g_z} =0.$$
Then $$\Sc(X,E)=\Sc(X-Z,E)+\Sc(X,E)_{(\g)}.$$
\end{theorem}
\pause
\begin{example}
$X=Z$ and we have a submersion $X \to W$ with all fibers - orbits.
\end{example}

%%%
\end{frame}

%%%




% \begin{frame}
%   \frametitle{Table of Contents}
%   \tableofcontents
%   Click TeX Refresh (F5) to update table of contents.
% \end{frame}

%\section{Introduction}

% \subsection{General Features}
%
% \begin{frame}
%   \frametitle{Presentation Outline}
%   \framesubtitle{Using sectioning commands}
%
%   You can use \texttt{\textbackslash section}
%   and \texttt{\textbackslash subsection} commands
%   to define outline of your presentation.
%
%   These commands must be appeared between frames.
%   Insert they manually in text source window.
%
%   To take correct outline under \TeX-Word click F5 key.
%
% \end{frame}
%
% \begin{frame}
%   \frametitle{Simple Overlays}
%   \framesubtitle{Using pause command}
%
%   The easiest way to create overlays is in using
%   the \texttt{\textbackslash pause} command.
%   Like following ...
%
%   \begin{itemize}
%     \item This item is before \texttt{pause} (look source window)\pause
%     \item If you don't see overlay read explanation below ...
%   \end{itemize}
%
%   Overlay expansion may be enabled/disabled
%   in dialog opened by `Insert/Document Properties' menu command.
%
%   `\textit{Overlays Mode=\textbf{Handout}}'
%    -- disables overlay expansion.
%
%   It is default setting of this template,
%   because this mode is convenient at
%   first stage of developing slides.
%
%   `\textit{Overlays Mode=\textbf{Beamer}}'
%    -- enables overlay expansion.
%
%   It is useful at final editing
%   for handling overlays and for generating
%   final slides in form of PDF file.
%
% \end{frame}
%
% \subsection{Advanced Features}
%
% \begin{frame}
%   \frametitle{Advanced Overlays}
%   \framesubtitle{Using more flexible overlay specifications}
%
%   Beamer includes more ways for handling
%   overlays\dots
%   \begin{itemize}
%   \item using the \texttt{pause} command:
%     \begin{itemize}
%     \item
%       First item.
%       \pause
%     \item
%       Second item.
%     \end{itemize}
%   \item
%     using overlay specifications:
%     \begin{itemize}
%     \item<3->
%       First item.
%     \item<4->
%       Second item.
%     \end{itemize}
%   \item
%     using the general \texttt{uncover} command:
%     \begin{itemize}
%       \uncover<5->{\item
%         First item.}
%       \uncover<6->{\item
%         Second item.}
%     \end{itemize}
%   \end{itemize}
%
%   To learn more about Beamer package
%   read original documentation.
%
% \end{frame}
%
% \section{Main Talk}
%
% \subsection{It is only beginning}
%
% \begin{frame}
%   \frametitle{Blank Frame}
%   Body.
% \end{frame}
%
% \section*{Conclusion}
%
% \begin{frame}
%   \frametitle{Conclusion}
%
%   Clicking right mouse button
%   (press+release without intermediate mouse moving)
%   displays context menu
%   which have two commands specific for beamer package:
%
%   \begin{itemize}
%   \item
%      \alert{Delete the Frame} -- removes current frame.
%   \item
%      \alert{Duplicate the Frame} -- easiest frame populating.
%   \end{itemize}
%
%   The button \includegraphics[height=0.75em]{slide.bmp}
%   at left edge of text
%   tool bar lets you insert blank slide in your presentation.
%
% \end{frame}


\end{document}




