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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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%\newcommand{\A}{\mathbb{A}}
\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}
% - Use the \inst{?} command only if the authors have different
%   affiliation.


\institute[Massachusetts Institute of Technology] % (optional, but mostly needed)
{Massachusetts Institute of Technology}
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% - Keep it simple, no one is interested in your street address.

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

%\subject{}
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% 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 Vladimir Drinfeld} \\$ $ \\ $ $ \\
\center{\url{http://math.mit.edu/~aizenr} }

\end{frame}


\begin{frame}
    \frametitle{\Rami{The Problem}}
%\thispagestyle{empty}

\pause
Let $F$ be a
%(non-Archimedean)
local field of characteristic $0$.
%(i.e. a finite extension of the field  of p-adic numbers $\Q_p$).
\pause

\begin{theorem}[Bernstein, Hrushovski-Kazhdan, Cluckers-Loeser]
Let $W$ be a finite-dimensional vector space over $F$ and $p$ be a polynomial on it.

\pause
Then
the Fourier transform of $|p|$ is smooth in an open dense set $U$.
\end{theorem}
\end{frame}


\begin{frame}
    \frametitle{\Rami{The Problem}}
%\thispagestyle{empty}

Let $F$ be a
%(non-Archimedean)
local field of characteristic $0$.
%(i.e. a finite extension of the field  of p-adic numbers $\Q_p$).
\begin{theorem} [Bernstein, Hrushovski-Kazhdan, Cluckers-Loeser, A.-Drinfeld]
Let $W$ be a finite-dimensional vector space over $F$ and $p$ be a polynomial on it.

Then
the Fourier transform of $|p|$ is smooth in an open dense set $U$.
\Rami{Moreover, $U$ is explicitly described by $p$ in algebro-geometric terms.}
\end{theorem}

%\begin{itemize}
%\item We do not require that $X$ be closed, but rather we ask for a certain regular behaviour of $\omega$ near the boundary of $X$ which will allow us to consider $|\omega|$ as a distribution on $W.$
%\item The main method of the proof is Hironaka desingularization theorem.
%\item Our method suits, in fact, both the Archimedean and the non-Archimedean case. However,  only the non-Archimedean case is discussed in the current version.
%\item Our method can give bounds on the open dense set $U$. These bounds are explicit in terms of resolution of singularities,  which, today, has rather explicit descriptions.
% \end{itemize}

\end{frame}



\begin{frame}
    \frametitle{The Archimedean Case}
%\thispagestyle{empty}
\pause
Let $X$ be a real (affine) algebraic manifold.
\pause

\begin{definition}
For a $D(X)$-module $M$, we set
$$SS(M):=Supp(gr(M)) \subset T^*X$$
\end{definition}
\pause

\begin{theorem} [Bernstein]
$$\dim(SS(M))\geq \dim(X)$$
\end{theorem}
\pause

\begin{theorem} [Malgrange, Kashiwara-Kawai-Sato, Gabber]
$SS(M)$ is coisotropic.
\end{theorem}
\end{frame}



\begin{frame}
    \frametitle{The Archimedean Case}

\begin{definition}
\begin{itemize}
\item
A $D(X)$-module $M$  is called holonomic if
$SS(M)$ is Lagrangian, or equivalently $\dim SS(M)=\dim(X)$.
\pause
\item For a distribution $\xi$ on $X$, we set
$SS(\xi):=SS(D(X) \xi)$
\pause
\item A distribution $\xi$  is called holonomic if
$SS(\xi)$ is Lagrangian, or equivalently $\dim SS(M)=\dim(X)$.
\end{itemize}
\end{definition}

\pause
\begin{theorem} [Bernstein]
\begin{itemize}
\item
If $\xi$ is holonomic then
$\xi$ is smooth in an open dense set.
\pause
\item
The class of holonomic distribution is closed under (proper) direct image, (submersive) inverse image, linear combinations \pause
and
  Fourier transform.
\end{itemize}
\end{theorem}
\end{frame}



\begin{frame}
    \frametitle{\Rami{A Stronger} Version}
\thispagestyle{empty}

\pause
\begin{theorem}
Let $X \subset W$ be a smooth closed algebraic subvariety. Let $\omega$ be an algebraic top differential form on it.
\pause
Let $|\omega|$ be the corresponding measure on $X$ and consider it as a distribution on $W$. %Fix an Haar measure on $W$.

\pause
Then the Fourier transform of $|\omega|$ is smooth in an open dense set.
\end{theorem}
\pause

\pause
\begin{theorem}
Let $Y$ be smooth algebraic variety and $X \subset Y \times W$.
\pause
Let $\omega$ be an algebraic top differential form on it.

\pause
Then the partial Fourier transform of $|\omega|$ w.r.t. $W$ is smooth in an open dense set.
\end{theorem}
\pause

In fact we do not require that $X$ be closed, but rather we ask for a certain regular behaviour of $\omega$ near the boundary of $X$ which will allow us to consider $|\omega|$ as a distribution on $W$.
\end{frame}



\begin{frame}
  \frametitle{The Wave Front Set}
\pause

Let $X$ be a smooth analytic variety. Let $\xi \in \Sc^*(X)$.
\pause
The wave front set of $\xi$ is a subset $WF(\xi) \subset T^*X$.
\pause

\begin{itemize}
\item   ${\Supp(\xi)} =
p_X(WF(\xi))=WF(\xi) \cap X$, where $p_X:T^*X \to X$ is the projection.
\pause
\item   $WF(\xi)$ is conical.
\pause
\item   $WF(\xi) \subset X$, iff $\xi$ is smooth.
\pause
\item Let $V$ be a linear space. Let $Z \subset V^*$ be a closed
subvariety, invariant with respect to homotheties. Let $\xi \in
\Sc^*(V)$. Suppose that $WF(\widehat{\xi}) \subset Z \times V$. Then
$WF(\xi) \subset V \times Z$.
\pause
\item \Rami{Behavior w.r.t.} group action:% (A.):
\\ Let an algebraic group $G$ act on $X$. Let $\xi \in \Sc^*(X)^{G, \chi}$. Then
$$WF(\xi) \subset \{(x,\phi) \in T^*X \, | \, \forall \alpha \in
\g, \quad \phi(\alpha(x)) =0\}=\bigcup_{y \in X} CN_{Gy}^X.$$
%\item Dierct image:\\ Let an algebraic group $G$ act on $X$. Let $\xi \in \Sc^*(X)^{G, \chi}$. Then
%$$WF(\xi) \subset \{(x,\phi) \in T^*X \, | \, \forall \alpha \in
%\g \quad \phi(\alpha(x)) =0\}.$$

\end{itemize}
\end{frame}

\begin{frame}
  \frametitle{The Wave Front Set}
\begin{itemize}
%\item Dierct image:\\ Let an algebraic group $G$ act on $X$. Let $\xi \in \Sc^*(X)^{G, \chi}$. Then
%$$WF(\xi) \subset \{(x,\phi) \in T^*X \, | \, \forall \alpha \in
%\g \quad \phi(\alpha(x)) =0\}.$$
\item Behavior w.r.t. inverse image:%  (H\"ormander, Heifetz):
\\
Let $p: Y \to X$ be a analytic submersion. Then $WF(p^*(\xi)) \subset p^*(WF(\xi))$.
\pause
\item Behavior w.r.t. direct image:%(Gabor, A.-Drinfeld):
\\
Let $p: X \to Y$ be a proper analytic submersion. Then $WF(p_*(\xi)) \subset p_*(WF(\xi))$.
\pause
\item  Weak integrability theorem:% (A.):
\\
Let $\xi \in \Sc^*(X)$. Then $WF(\xi)$ is
weakly coisotropic.
\end{itemize}

\end{frame}


\begin{frame}
  \frametitle{WF-Holonomic Distributions}
  \thispagestyle{empty}

\pause
\begin{definition}
We call a distribution $\xi$ on an analytic manifold $X$ "WF-holonomic" if the following equivalent conditions are satisfed:
\pause
\begin{itemize}
%\item the zarizki cloger of $WF()$ is isotropic. \label{5}
\item $WF(\xi)$ is included in an isotropic subvariety
of $T^*(X)$.
\pause
\item $WF(\xi)$ is included in a Lagrangian subvariety
of $T^*(X)$.
\pause
\item  There is a finite collection of smooth (locally closed) subvarieties $A_i \subset X$ such that $$WF(\xi) \subset \bigcup_i \overline{CN_{A_i}^X}$$
\pause
\item  There is a finite collection of smooth (locally closed) subvarieties $A_i \subset X$ such that $$WF(\xi) \subset \bigcup_i {CN_{A_i}^X}$$
\end{itemize}
\end{definition}

\end{frame}

\begin{frame}
    \frametitle{WF Relative Version}
%\thispagestyle{empty}


\pause
\Rami{
\begin{proposition}
A WF-holonomic distribution is smooth in an open dense set.
\end{proposition}
}
\pause
\begin{theorem} [A.-Drinfeld]
Let $Y$ be smooth algebraic variety and $X \subset Y \times W$.
\pause
Let $\omega$ be an algebraic top differential form on it.
\pause
Then the partial Fourier transform of $|\omega|$ w.r.t. $W$ is WF-holonomic.
\end{theorem}

\end{frame}

\begin{frame}
    \frametitle{Hironaka's Theorem}
%\thispagestyle{empty}

\pause
\begin{theorem}[Hironaka]
Let $X$ be an algebraic variety. Then there exists a resolution of singularities $p:Y \to X$, i.e. a proper surjective map which is isomorphism on an open dense set.

\pause
Moreover, let $D \subset X$ be a divisor. Then we can take $p$ s.t. $p^{-1}(D)$ will be a divisor with normal crossings.
\end{theorem}
\end{frame}

\begin{frame}
    \frametitle{WF Graph Version}
%\thispagestyle{empty}

\pause
\begin{theorem}
Let
\begin{itemize}
\item
$Y$ be smooth analytic variety and $f:Y\to \P^n$ be an analytic map.
\pause
\item
$Z_1:=f^{-1}(\P^n-\A^n)$ and $U=Y-Z_1$.
\pause
\item
 $i:U \to Graph(f) \to Y \times \A^n$.
\pause
\item
 $\omega$ be a meromorphic top differential form on $Y$ which is regular in $U$, and $Z_2$ be the zero set of $\omega$.
\end{itemize}
\pause
Assume that $Z_1 \cup Z_2$ is a divisor with normal crossings.

\pause
Then the partial Fourier transform of $i_* \left|(\omega|_U) \right|$ w.r.t. $\A^n$ is WF-holonomic.
\end{theorem}

\end{frame}



\begin{frame}
    \frametitle{Baby WF Graph Version}
%\thispagestyle{empty}

\pause
\begin{theorem}
Let $Y$ be smooth analytic variety and $f$ be a meromorphic function on $Y$.
Let $Z_1:=f^{-1}(\infty)$ and let $U=Y-Z_1$. Let $i:U \to Graph(f) \to Y \times F$.
\pause
Let $\omega$ be a meromorphic top differential form on $Y$ which is regular in $U$. Let $Z_2$ be the zero set of $\omega$.
\pause
Assume that $Z_1 \cup Z_2$ is a divisor with normal crossings.

\pause
Then the partial Fourier transform of $i_* \left|(\omega|_U) \right|$ w.r.t. $F$ is WF-holonomic.
\end{theorem}
\pause

Local model: \pause
 $f$ and $\omega$ are monomials.


\end{frame}

\end{document}
