\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{motivic integration}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration theory - contents]]

\hypertarget{motivic_cohomology}{}\paragraph*{{Motivic cohomology}}\label{motivic_cohomology}

[[!include motivic cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

What is called \emph{motivic integration} is an upgrade of [[p-adic integration]] to a geometric integration theory obtained by replacing the [[p-adic integers]] by a [[formal power series ring]] over the [[complex numbers]], and more generally by henselian discretely valued fields of residual characteristic zero.

\hypertarget{history}{}\subsection*{{History}}\label{history}

Motivic integration was introduced in the talk of [[Maxim Kontsevich]] at Orsay in 1995 in order to prove that [[Hodge numbers]] of [[Calabi-Yau manifolds]] are [[birational map|birational]] invariants. This talk also dealt with [[orbifold cohomology]] as well as 2 related papers of [[Lev Borisov]]. The orbifold cohomology has been continued by Weimin Chen, [[Yongbin Ruan]] and collaborators, and later also by algebraic geometers Abramovich, Vistoli, and others. From physical side a pioneer of both subjects is also Batyrev.

Later, more general framework of motivic integration in model theory has been put forward by Denef and Loeser, partly based on Denef's work on $p$-adic integration. More recent work using model theoretical approach is by Hrushovski and Kazhdan.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[p-adic integration]]


\item [[motivic cohomology]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

A textbook account is in

\begin{itemize}%
\item Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag, \emph{Motivic integration}, Progress in Mathematics 325 Birkhaeuser 2018 (\href{https://doi.org/10.1007/978-1-4939-7887-8}{doi:10.1007/978-1-4939-7887-8})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia \emph{\href{https://en.wikipedia.org/wiki/Motivic_integration}{Motivic integration}}

\end{itemize}
Original articles include the following:

\begin{itemize}%
\item Jan Denef, [[François Loeser]], \emph{Definable sets, motives and $p$-adic integrals}, J. Amer. Math. Soc. \textbf{14} (2001), no. 2, 429--469, \href{http://dx.doi.org/10.1090/S0894-0347-00-00360-X}{doi}


\item Jan Denef, Fran\c{c}ois Loeser, \emph{Motivic integration and the Grothendieck group of pseudo-finite fields} Proc. ICM, Vol. II (Beijing, 2002), 13--23, Higher Ed. Press, Beijing, 2002.


\item R. Cluckers, F. Loeser, \emph{Constructible motivic functions and motivic integration}, Invent. Math. \textbf{173} (2008), 23--121 \href{http://arxiv.org/abs/math/0410203}{math.AG/0410203}


\item Jan Denef, Fran\c{c}ois Loeser, \emph{Germs of arcs on singular algebraic varieties and motivic integration}, Invent. Math. \textbf{135} (1999), no. 1, 201--232.


\item D. Abramovich, M. Mari\~n{}o, M. Thaddeus, R. Vakil, \emph{Enumerative invariants in algebraic geometry and string theory}, Lectures from the C.I.M.E. Summer School, Cetraro, June 6--11, 2005. Edited by Kai Behrend and Marco Manetti. LNIM 1947, Springer 2008. x+201 pp.


\item Manuel Blickle, \emph{A short course on geometric motivic integration}, \href{http://arxiv.org/abs/math/0507404}{math.AG/0507404}


\item [[Ehud Hrushovski]], [[David Kazhdan]], \emph{Motivic Poisson summation}, \href{http://arxiv.org/abs/0902.0845}{arxiv/0902.0845}


\item [[Ehud Hrushovski]], [[David Kazhdan]], \emph{The value ring of geometric motivic integration and the Iwahori Hecke algebra of $SL_2$}, \href{http://arxiv.org/abs/math/0609115}{math.LO/0609115}; \emph{Integration in valued fields}, in Algebraic geometry and number theory, 261--405, Progress. Math. \textbf{253}, Birkh\"a{}user Boston, \href{http://math.huji.ac.il/~ehud/papers/intv-060428.pdf}{pdf}


\item Julia Gordon, Yoav Yaffe, \emph{An overview of arithmetic motivic integration}, \href{http://arxiv.org/abs/0811.2160}{arxiv/0811.2160}


\item Thomas C. Hales, \emph{What is motivic measure?}, \href{http://arxiv.org/abs/math/0312229}{math.LO/0312229}


\item [[David Kazhdan]], \emph{Lecture notes in motivic integration}, with intro to logic and model theory, \href{http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/b.pdf}{pdf}


\item R. Cluckers, J. Nicaise, J. Sebag (Editors), \emph{Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry}, 2 vols. London Mathematical Society Lecture Note Series \textbf{383}, \textbf{384}


\item [[Raf Cluckers]], \emph{Motivic integration for dummies}, \href{http://wis.kuleuven.be/algebra/Raf/MForD.pdf}{pdf}, A course on motivic integration \href{http://wis.kuleuven.be/algebra/Raf/LaRoche3Cluckers.pdf}{I}, \href{http://wis.kuleuven.be/algebra/Raf/LaRoche3Cluckers.pdf}{II}, \href{http://wis.kuleuven.be/algebra/Raf/LaRoche3Cluckers.pdf}{III}


\item Raf Cluckers, Julia Gordon, Immanuel Halupczok, \emph{Motivic functions, integrability, and uniform in p bounds for orbital integrals}, \href{http://arxiv.org/abs/1309.0594}{arxiv/1309.0594}


\item [[Lou van den Dries]], \emph{Lectures on Motivic Integration} , Ms. University of Illinois at Urbana-Champaign. (\href{http://www.math.uiuc.edu/~vddries/mo.dvi}{dvi})


\item Julien Sebag, \emph{Int\'e{}gration motivique sur les sch\'e{}mas formels}, Bull. Soc. Math. France 132 (2004), no. 1, 1--54, \href{http://www.ams.org/mathscinet-getitem?mr=2075915}{MR2005e:14017}


\item Takehiko Yasuda, \emph{Motivic Integration over Deligne-Mumford Stacks} , arXiv.0312115 (2004). (\href{http://arxiv.org/pdf/math/0312115v5.pdf}{pdf})


\item M. Larsen, [[Valery Lunts]], \emph{Motivic measures and stable birational geometry}, Mosc. Math. J. 3 (2003), no. 1, 85--95, 259, \href{http://arxiv.org/abs/math/0110255}{math.AG/0110255}, \href{http://www.ams.org/mathscinet-getitem?mr=1996804}{MR2005a:14026}, \href{http://www.ams.org/distribution/mmj/vol3-1-2003/abst3-1-2003.html#larsen-lunts_abstract}{journal}; \emph{Rationality criteria for motivic zeta functions}, Compos. Math. \textbf{140} (2004), no. 6, 1537--1560, \href{http://arxiv.org/abs/math/0212158}{math.AG/0212158}


\item [[Alexei Bondal]]. M. Larsen, [[Valery Lunts]], \emph{Grothendieck ring of pretriangulated categories}, Int. Math. Res. Not. 2004, no. 29, 1461--1495, \href{http://arxiv.org/abs/math/0401009}{math.AG/0401009}


\item Emmanuel Bultot, \emph{Motivic integration and logarithmic geometry}, PhD thesis \href{http://arxiv.org/abs/1505.05688}{arxiv/1505.05688}


\item Karen Smith, \emph{Motivic integration}, (\href{http://www.msri.org/ext/shortermotivic2.pdf}{pdf})



\end{itemize}
[[!redirects motivic measure]]



\end{document}