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  \begin{document}
 {\bf
\title{\Huge \textcolor{blue}{Abel Prize 2004: \\
The Atiyah--Singer index theorem}}
\author{{\Large Klaas Landsman} \\  \mbox{} \hfill \\
\tc{red}{Radboud Universiteit Nijmegen}\\ and \\  \tc{red}{Fellowship of Geometry and Quantum Theory} \\ \mbox{} \hfill \\ 
\texttt{landsman@math.ru.nl} \\   \mbox{} \hfill \\
NMC 2006, 27 March, Delft}\date{}
 \maketitle {\Large

\newpage
\ig[width=\textwidth]{atiyah_singer.jpg}  
 \tc{red}{
$$\ind(D)=\int_{T^*M} \mathrm{Ch}([\sigma(D)]) \mathrm{Td}(T^*_{\mathbb{C}}M) $$}\newpage
 {\Huge\noindent\textcolor{blue}{Why is the index theorem famous?}}\medskip
 
\begin{itemize}
\item Relates algebra, analysis, geometry and topology:
\begin{trivlist}
\item \tc{red}{l.h.s.\ $\sim$  functional analysis (linear algebra in $d=\infty$)
\item r.h.s.\ $\sim$ differential geometry and algebraic topology}
\end{trivlist}
\item Synthesis and extension of many known deep results:
\begin{trivlist}
\item  \tc{red}{Gau\ss--Bonnet, Riemann--Roch, de Rham,\ldots}
\end{trivlist}
\item Very hard to prove (yet has many different proofs)
\item Created its own research field: index theory ($\rightarrow$ NCG)
\item Unexpected applications to quantum physics:
\begin{trivlist}
\item \tc{red}{Anomalies in quantum field theory $\rightarrow$ baryogenesis 
\item Quantization theory (Raoul Bott, 1923--2005)}
\end{trivlist}
\end{itemize}\newpage
\newpage  {\Huge\noindent\textcolor{blue}{The index in linear algebra}} \bigskip

\noindent A linear map $L:V\raw W$ defines  an equation $Lv=w$
\medskip

 \tc{blue}{Are there any solutions?} (take inner product on $W$)
 \begin{trivlist}\item
Yes, iff  $w\in \mathrm{ran}(L)=  \mathrm{ran}(L)^{\perp\perp}\subset W$ or: $w\in \mathrm{coker}(L)^{\perp}$\item with 
 $\mathrm{coker}(L):= \mathrm{ran}(L)^{\perp}$; w.r.t.\ a basis $(e_i)$ of 
$\mathrm{coker}(L)$, \item this  gives 
 \tc{red}{$\dim(\mathrm{coker}(L))$}  conditions $(w,e_i)=0$ on $w$  
\end{trivlist}

\tc{blue}{What is the dimension of the solution space  $S$?} 
 \begin{trivlist}\item
$\dim(S)=$ \tc{red}{ $\dim(\mathrm{ker}(L))$}: $L(v+v')=w$ if $Lv=w$ and $Lv'=0$
\end{trivlist}

 \tc{blue}{Discontinuous dependence on $L$:}  if $V=W$, then
 $$\dim(\mathrm{ker}(\varepsilon\cdot 1  ))=\dim(\mathrm{coker}(\varepsilon\cdot 1))=\tc{red}{0}
  \:  \mathrm{if}\:  \tc{red}{\varepsilon\neq 0}\: 
 \mathrm{but}\: \tc{blue}{\dim(V)}\: \mathrm{if}\:
 \tc{blue}{\varepsilon=0}$$
 
\tc{blue}{Stable quantity:} \tc{red}{$\ind(L):=\dim(\mathrm{ker}(L))-\dim(\mathrm{coker}(L))$}
\newpage  {\Huge\noindent\textcolor{blue}{The baby index theorem ($\dim(\cdot)<\infty$)}} 

\tc{red}{$$\ind(L)=\dim(V)-\dim(W)$$}
\begin{enumerate}
\item $\ind(L)$ is independent of $L$
\item depends on $V$ and $W$ only through topological \\ invariants $\dim(V)$ and $\dim(W)$ (L.E.J. Brouwer)
\item The index of $L:V\raw V$ is 0
\end{enumerate}

\noindent \tc{blue}{The  index of an infinite-dimensional linear operator $L$:}
\begin{itemize}
\item may depend on $L$ itself as well (again  topologically)
\item may be nonzero even if $V=W$
\item may even be infinite; hence require \tc{red}{$\dim(\mathrm{ker}(L))<\infty$} and \tc{red}{$\dim(\mathrm{coker}(L))<\infty$} i.e.\ $L$ is   `almost invertible' 
\end{itemize}
 {\Huge\noindent\textcolor{blue}{Elliptic differential operators}} \bigskip
 
 \begin{description}
\item[] Example: \tc{red}{$D_1=\frac{\partial}{\partial x} -i \frac{\partial}{\partial y}$} on $V=W=L^2(\R^2)$
\item[] $f\in\ker(D_1)$ $\Rightarrow$ $f=f(x+iy)$ $\Rightarrow$ $\dim(\ker(D_1))=0$ (Liouville)
\item[] Also:  $\dim(\coker(D_1))=0$ $\Rightarrow$ \tc{red}{$\ind(D_1)=0-0=0$} (R--R)
\item[] Not an example: \tc{red}{$D_2=\frac{\partial}{\partial x} - \frac{\partial}{\partial y}$} on $L^2(\R^2)$ (has no index)
\item[] Better example: \tc{red}{$D_3=\frac{d}{d x} +x$} on $L^2(\R)$
\item[]
$f\in\ker(D_3)$ $\Rightarrow$ $f\sim \exp(-x^2/2)$  $\Rightarrow$ $\dim(\ker(D_3))=1$ 
\item[]
$f\in\coker(D_3)$ $\Rightarrow$ $f\sim \exp(+x^2/2)$  $\Rightarrow$ $\dim(\ker(D_3))=0$ 
\item[] Hence \tc{red}{$\ind(D_3)=1-0=1$}
\end{description}

 Atiyah--Singer  is about  \tc{red}{elliptic} (pseudo) differential\\  operators (like $D_1$ and $D_3$) on \tc{red}{compact} spaces
 \newpage
  {\Huge\noindent\textcolor{blue}{Dirac operators}} \bigskip
  
  Two ways to probe a space with elliptic operators:
  \begin{itemize}
\item structureless particle $\raw$  \tc{red}{Riemannian geometry} \&\ \\ \tc{red}{Laplace operator} $\Delta$ (e.g.\ $\Delta=\partial/\partial x^2 + \partial/\partial y^2$ on $L^2(\R^2)$)
\item spinning particle with internal structure $E$ (vector bundle) $\raw$  `\tc{red}{spin geometry}' \&\
\tc{red}{Dirac operator} $\DS_E$ 
\end{itemize}
e.g. on $\R^2$ with $z=x+iy$: $\DS=\left( \begin{array}{cc}
0&  -\frac{\partial}{\partial z}\\ \frac{\partial}{\partial\overline{z}}  & 0
\end{array}
\right) $ on $L^2(\R^2)\otimes \C^2$\bigskip

  
\noindent {\small
\tc{blue}{ ``The geometrical significance of spinors is still very mysterious. (\ldots) When Singer and I were investigating these questions we `rediscovered' for ourselves the Dirac operator. Had we been better educated in physics, or had there been the kind of dialogue with physicists that is now common, we would have got there much sooner"} (Michael Atiyah, 1998)}

  \newpage
  {\Huge\noindent\textcolor{blue}{Index theorem for  Dirac operators}} \bigskip

\tc{red}{$$\ind(\DS_E)=\int_M \hat{A}(M)\mathrm{ch}(E)
$$}
\begin{enumerate}
\item $\ind(\DS_E)$ is independent of $\DS_E$
\item depends on space $M$ and internal structure $E$ only through topological  invariants 
$\hat{A}(M)$ and $\mathrm{ch}(E)$
\item The index of $\DS_E$ may be nonzero!
\end{enumerate}
\tc{red}{Two ways to read the index formula:}
\begin{itemize}
\item \tc{blue}{`topological' r.h.s.\ computes `analytical' l.h.s.}
\item \tc{blue}{l.h.s.\ explains integrality of topological  r.h.s.}
\end{itemize}

e.g.\ Gau\ss--Bonnet: \tc{red}{$(\# \Delta)-(\#\partial) +(\# V)=(1/2\pi) \int_{\Sigma} K$}
\newpage
 {\Huge\noindent\textcolor{blue}{Application 1: baryogenesis}} \bigskip

 {\small ``The index theorem created matter in the early Universe" (Ed Witten, 2004)}\smallskip
 
 \begin{description}
\item[ ] \tc{red}{Baryons} $B$ (protons, neutrons)  have \tc{red}{antiparticles} $\overline{B}$
\item[ ] Today: $(n_B-n_{\overline{B}})/(n_B+n_{\overline{B}})\simeq 1$ but $<10^{-9}$ after Big Bang
\item[ ] Naive conservation law: $\frac{d}{dt} (n_B-n_{\overline{B}}) =0$ \tc{red}{$\Rightarrow$}
\item[ ] \tc{red}{Problem:} how did baryons wipe out antibaryons?
\end{description}\medskip

 Solution due to  Sakharov (1967), 't Hooft (1976), \ldots,:
 \tc{red}{$$ \frac{d}{dt} (n_B-n_{\overline{B}}) \sim \ind(\DS_B)\neq 0$$}
 
 \noindent{\small ``However, now a consensus has been reached that the effect is far too small.
  (\ldots) However, all of the above may be wrong (\ldots)" (A. Dolgov, 1997)}
  \medskip
  
  \tc{blue}{Viability depends on future experimental  data!}
 \newpage
 {\Huge\noindent\textcolor{blue}{Application 2: what is quantization?}} \bigskip

 \begin{description}
\item[] Traditionally (Heisenberg, Dirac, von Neumann, $\sim$1930):
\item[ ] \tc{red}{(i) phase space (symplectic manifold) $\raw$ Hilbert space}
\item[ ] \tc{red}{(ii) functions on phase space $\raw$  linear operators}
\end{description}
\begin{description}
\item[] \tc{blue}{Abstraction 1} (Guillemin \&\ Sternberg, 1982): 
\item[    ] \tc{red}{Quantization is equivalence class of unitary $G$-modules}
\item[] \tc{blue}{Abstraction 2} (Bott, $\sim$ 1990):   \tc{red}{Quantization is 
\item[ ] `$G$-index' of Dirac operator $\DS_L$ on phase space}
\item[]  `Dutch school': functoriality of Bott's definition \tc{red}{$\Rightarrow$}
\item[] 
 \tc{blue}{Abstraction 3} (2005): \tc{red}{Quantization is a functor}
\end{description}
\medskip 

But: \tc{blue}{mathematical precision $\sim$ 1/physical relevance!}


}}\end{document}