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\begin{document}


%\def\Bbb{\bf }
\title{Ideas for the Simons Wall Project}
\author{Claude LeBrun\\
Department of Mathematics\
\\Stony
 Brook}
\maketitle

\pagebreak 
%hyperbolic surface
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\vfill

{\tt 

\noindent 
Explanation:

\bigskip



\noindent 
This diagram  depicts the uniformization of a surface of genus $2$.
It shows that such such a surface can be given a metric of constant Gauss
curvature by displaying it as a quotient of  hyperbolic $2$-space by a group of
isometries. }


\pagebreak 

{\Large


$$\Dirac :  \Gamma ({\mathbb S}_+\otimes E) \to \Gamma ({\mathbb S}_-\otimes E)$$

$$\dim \ker  (\Dirac) - \dim \ker  (\Dirac^*)  = \int_{M^{4k}} \hat{A}(M) \smile ch (E)$$
}



\pagebreak 

{\Large


$$\Dirac :  \Gamma ({\mathbb S}_+\otimes E) \to \Gamma ({\mathbb S}_-\otimes E)$$

$$\dim \ker  (\Dirac) - \dim \ker  (\Dirac^*)  = \int_{M^{4k}} \hat{A}(M) \smile ch (E)$$

$$\hat{A} = 1-\frac{p_1}{24} + \frac{7p_1^2-4p_2}{5760} + \cdots$$

$$ch = \mbox{rank} +\frac{c_1}{2} + \frac{c_1^2-2c_2}{2} +\cdots$$

}



\vfill

{\tt 

\noindent 
Explanation:

\bigskip



\noindent 
This is the Atiyah-Singer index formula for the twisted Dirac operator, 
where $E$ is a complex  
vector bundle over 
a smooth compact  spin manifold
of dimension $4k$. The additional formul{\ae}
on this page give more information on the terms
involved. This arguably makes the display too busy,
however.}



\pagebreak 

{\Large

$$
\chi (M^{2n})= \frac{1}{(8\pi)^n n!}
\int_{M}\underbrace{R^{ij}_{ab}\cdots R^{k\ell}_{cd}}_n
\varepsilon^{ab\cdots cd}
\varepsilon_{ij\cdots k\ell}
~d\mu 
$$

}

\pagebreak 

{\Large

$$
\chi (M^{2n})= \frac{1}{(8\pi)^n n!}
\int_{M}\underbrace{R^{ij}_{ab}\cdots R^{k\ell}_{cd}}_n
\varepsilon^{ab\cdots cd}
\varepsilon_{ij\cdots k\ell}
~d\mu 
$$

}

\vfill

{\tt 

\noindent 
Explanation:

\bigskip



\noindent 
This is the generalized Gauss-Bonnnet theorem. 
It expresses the Euler characteristic of any smooth 
compact even-dimensional manifold $M$ as a  curvature integral
that can be computed using any 
Riemannian metric $g$ on $M$.}


\end{document}
