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% This file is a solution template for:

% - Giving a talk on some subject.
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% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
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% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
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% notice.


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%\title % (optional, use only with long paper titles)
%{Fourier transform of algebraic measures}
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%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
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{A. Aizenbud}
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{Weizmann Institute of Science}
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%{August 14, 2008}

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\title % (optional, use only with long paper titles)
%[Analysis, Arithmetic, Geometry and Algebra]
%{A Meeting Point for Analysis, Arithmetic, Geometry and Algebra}
[WF-holonomicity of constructible distributions]%
{WF-holonomicity of constructible distributions on non-Archimedean local fields}
%
%%\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
\begin{document}


\begin{frame}
\myTP
\center{Joint with Raf Cluckers}

$ $

$ $

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

\end{frame}




\begin{frame}
\myFT{Distributions}

\myP

\begin{defn}
\begin{itemize}
\myIt
Test functions -- smooth compactly supported functions / Schwartz functions.
\myIt
Distributions  -- functionals on  test functions.
\end{itemize}
\end{defn}
\myP


\begin{examples}
Delta function $\delta$, its derivative $\delta'$,  any locally $L^1$ function, $|p| ^\lambda$
\end{examples}
\myP

\textbf{Operations with distributions:}

\begin{itemize}
\myIt
pullback
\myIt
push forward
\myIt
Fourier transform
\myIt
Derivation
\myIt
Algebraic operations: $+,\cdot,\boxtimes$
\end{itemize}

\end{frame}




\begin{frame}
\myFT{The Archimedean case}
%\myTPSE
\myP
\begin{defn}
Holonomic distributions --  distributions that satisfy lots of PDE:\myP


Let $\xi \in \Sc^*(V)$ be a distribution on vector space. $\xi$ is holonomic iff \myP
$$ \dim Char(\xi):=\dim(Zeros(\{Sym(D)|D\xi=0\}))=\dim V.$$
\end{defn}
\myP

\begin{theorem}[Bernstein  $\sim$1970]
\begin{itemize}
\myIt
the class of holonomic distributions is closed under all of  the operations above whenever these are defined
\myIt $\dim Char(\xi) \geq \dim V$.
\end{itemize}
\end{theorem}
\myP
 \begin{theorem} [Kashiwara-Kawai-Sato, Malgrange, Gaber $\sim$1980]
$Char(\xi)$  is co-isotropic.
\end{theorem}
\myP
%\small{\quotebox{
%$\text{``}$All the distributions which appear in nature are holonomic$\text{''}$
\enquote{All the distributions which appear in nature are holonomic.}
%}}
\end{frame}


\begin{frame}
 \myFT{Wave front set}
\myP
\begin{observation}
$\xi$ is smooth iff $\hat \xi$ is rapidly decaying.
\end{observation}
\myP
\begin{defn}
Let $\xi \in \Sc^*(V)$ is a distribution on vector space.
\begin{itemize}
\myIt
We say that $\xi$ is smooth at point $x$ and direction $v$ if  $\widehat {\rho\xi}$ is rapidly decaying at direction $v$, where $ \rho $ is  a cut-off function of a small enough neighborhood of $x$
\myIt $WF(\xi) = \{(x,v)\in  T^*V|\xi \text{ is not smooth at  } (x,v)\}$.
\end{itemize}
\end{defn}
\myP


\begin{theorem}[H\"{o}rmander  $\sim$1980]
\begin{itemize}
\myIt  $WF(\xi)$ is invariant w.r.t. diffeomorphisms.
\myIt $WF(\xi) \subset Char(\xi)$.%:= Zeros(\{Sym(d)|D\xi=0\} $
\end{itemize}
\end{theorem}

 \end{frame}


\begin{frame}
\myTPSE
 \myFT{p-adic numbers}
%\begin{footnotesize}
{\fontsize{10.35}{10.35} \selectfont

\begin{defn}
\begin{itemize}
\myIt
p-adic numbers are \enquote{numbers} who have a \enquote{p-cimal} presentation which is finite after the  \enquote{p-cimal point} and possibly infinite before it.
\end{itemize}
Alternatively:
\begin{itemize}
\myIt
The field of p-adic numbers $\Q_p$ is the completion of $\Q$ w.r.t. the p-adic norm:
$$\left|p^k\frac{m}{n}\right| =p^{-k}, \quad \text{where: } gcd(p,n)= gcd (p,m)=1.$$
\end{itemize}
\end{defn}

 \begin{itemize}
\myIt
  Although we consider p-adic numbers as arguments, the values of our functions are always complex.
\myIt
 Smooth functions on $\Q_p$ are locally constant functions.
\myIt
 Rapidly decaying functions are functions with compact support
\myIt
 This gives us the notion of distribution.
\myIt
 Instead of using the periodic exponent   $e^{ix}$ one uses a fixed additive character $\psi(x)$.
\myIt
 This gives us the notion of Fourier transform and wave front set.
\myIt[ \frownie{}]
No action of differential operators on distributions.
 \end{itemize}
%{\large  \frownie{}}   No action of differential operators on distributions
%
}
%\end{footnotesize}

\end{frame}




\begin{frame}
 \myFT{Wave front holonomicity}
\myP
\begin{theorem} [A. 2008]
$WF(\xi)$ includes Lagrangian, in particular $\dim   WF(\xi) \geq \dim V$.
\end{theorem}
\myP
\begin{defn}
$\xi$ is WF-holonomic if  $WF(\xi)$ is isotropic. In particular  $\dim WF(\xi)=\dim V$.
\end{defn}
\myP
\begin{theorem} [A.-Drinfeld 2011]
\begin{itemize}
\myIt  Many distributions with algebraic description (and their Fourier transforms)  are  WF-holonomic.
\myIt WF-holonomicity is stable under proper push-forward and submersive pull-back.
\end{itemize}
\end{theorem}
\myP
%:-(
{\large  \frownie{}}   WF-holonomicity is not  stable under Fourier transform.


\end{frame}


\begin{frame}
 \myFT{Constructible functions}
\myTPSE
\myP
\textbf{
Functions that have a nice formula.
}
\myP
{\small
\begin{examples}
\begin{itemize}
\myIt
Absolute value of a rational function.
\myIt
Valuation (log of the absolute value) of a rational function.
\myIt
$\psi$ composed with a rational function.
\myIt Characteristic function of a ball.
\end{itemize}
\end{examples}
\begin{defn}
The algebra of constructible functions is the algebra generated by (generalizations of) the above examples.
\end{defn}
\myP
 non-example: $\frac{1}{log}$.
\myP
\begin{theorem}[Clukers-Loeser 2005]
The class of  constructible functions is closed under the above operations, whenever defined.
\end{theorem}
\myP
\enquote{All the functions which appear in nature are constructible}
}
\end{frame}



\begin{frame}
\myFT{(p-adic) Wavelet transform}
%{\color{red}{
\myP
\begin{defn}
Let $ F $ be  a p-adic (more generally non-Archimedean local) field. \myP
Define:\myP
$$WL:\Sc^*(V)\to C^\infty( V \times F^ \times )  $$\myP
$$ WL (\xi)(a,b):=\langle \xi ,1_{B(a,|b|)}\rangle  $$\myP
\end{defn}
%}}
It is easy to see that WL is 1-1.
\end{frame}


\begin{frame}
\myFT{Constructible distributions}
\myP
 \begin{defn}
$\xi$  is constructible iff $WL(\xi)$ is constructible.
\end{defn}

\myP


\begin{theorem}[Cluckers-Halupczok-Loeser-Raibaut, 2018]
\begin{itemize}
\myIt
The class of constructible distributions is closed under all the above operations, whenever defined.
\myIt Constructible distributions are smooth almost everywhere.
\end{itemize}
\end{theorem}

\myP
\enquote{All the distributions which appear in nature are constructible}

\end{frame}




\begin{frame}
\myFT{Main Result}
\myP

\begin{theorem}[A.-Cluckers 2019]
Constructible distributions are  WF-holonomic .
\end{theorem}
\myP
%{\color{violet}{
%\textbf{Main ingredients of the proof:}
\begin{proof}[Main ingredients of the proof]
\begin{itemize}
%{\color{violet}{
\myIt {\it Regularization:} a constructible distribution can be extended from an open set.

\myIt {\it Resolution of singularities in the constructible (in fact, definable) setting.}

\myIt {\it Key-Lemma:} a smooth constructible function on an open set can be extended to an holonomic constructible distribution.
%}
\end{itemize}
\end{proof}
%}}
\end{frame}




\begin{frame}
%\myFT{Idea of the proof}
\begin{proof}[Idea of the proof]

\begin{itemize}
%\color{violet}{
\myIt $\xi|_U$ is smooth for open dense $U$.
\myIt  Extend   $\xi|_U$  to an holonomic  constructible distribution $\xi'$.
\myIt  Let $\eta= \xi' - \xi$ . We have $\dim supp( \eta )<\dim V$.
\myIt  Resolve $Z=supp( \eta )$ by a smooth manifold:   $$p:M\to Z$$.
\myIt  Let $Z'\subset Z$ open dense s.t. $p^{-1}(Z')\cong Z'$.
\myIt  Extend $p^*(\eta|_{ Z' })$ to constructible distribution $\mu$ on  $M$.
\myIt  By the induction assumption, $\mu$ is WF-holonomic.
\myIt  Thus $p_*(\mu)$ is constructible  WF-holonomic.
\myIt  By the induction assumption   $ p_*(\mu) -\eta$ is  WF-holonomic.
%}
\end{itemize}
\end{proof}
\end{frame}




\begin{frame}
%\myFT{Idea of the Proof the Key Lemma}
\myFT{The Key Lemma}
%\myTPSE
%Idea of the proof of the Key Lemma
\begin{small}
\begin{keylemma}
Let $f$ be a constructible function on an open (definable) set $U\subset V$. Then $f$ can be extended to a constructible WF-holonomic distribution on  $V$.
\end{keylemma}
\myP
\begin{proof}[Idea of the Proof]
\begin{itemize}
\itemsep0em 
%\color{violet}{
\myIt WLOG we can assume that the function $f$ has the form:
$\psi(p_1)|p_2|val(p_3)$
\myIt  Using resolution of singularities we may assume that U is the complement of the coordinate hyper planes and  $p_i=u_i m_i$, where $u_i$ are units and $m_i$ are monomials.
\myIt[{  \frownie{}}]   While $u_2$ and $u_3$ can be ignored, $u_1$ cannot.
\myIt  Instead we can swallow it in $m_1$.
\myIt  Now we prove the Key lemma for the complement of the origin.
\myIt  We are using an inductive assumption both about the Key lemma and the main theorem.
\myIt  Adding 1 point does not affect WF-holonomicity.
%}
\end{itemize}
\end{proof}
\end{small}

\end{frame}



\end{document}

