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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 


Uniformization of a surface of genus $2$:
\bigskip 

%hyperbolic surface
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\circulararc -45 degrees from  0 -59  center at 0 -100
\circulararc 45 degrees from   -59 0 center at  -100 0
\circulararc -45 degrees from  -59   0 center at  -100 0
\circulararc 45 degrees from  59 0  center at  100 0
\circulararc -45 degrees from  59 0 center at  100 0
\circulararc 45 degrees from  42 -42  center at  71 -71
\circulararc -45 degrees from  42 -42 center at  71 -71
\circulararc 45 degrees from  42 42  center at  71 71
\circulararc -45 degrees from  42 42 center at  71 71
\circulararc 45 degrees from  -42 -42  center at  -71 -71
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center at 120 33
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center at 120 29
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center at 205 33
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\end{center}

\pagebreak 


Index theorem for coupled Dirac operator: 

{\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 

Generalized Gauss-Bonnnet theorem:




\bigskip
{\Large

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

}

\vspace{1in}


Classical Gauss-Bonnnet theorem:




\bigskip
{\Large

$$
\chi (M^{2})= \frac{1}{2\pi}
\int_{M}K 
~dA 
$$
}



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\ellipticalarc axes ratio 5:2  280 degrees from 150 40
center at 120 30
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center at 205 30
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center at 205 30
\ellipticalarc axes ratio 5:2  -280 degrees from 260 40
center at 290 30
\ellipticalarc axes ratio 4:1 -180 degrees from 135 33
center at 120 33
\ellipticalarc axes ratio 4:1 145 degrees from 130 30
center at 120 29
\ellipticalarc axes ratio 4:1 180 degrees from 190 33
center at 205 33
\ellipticalarc axes ratio 4:1 -145 degrees from 195 30
center at 205 29
\ellipticalarc axes ratio 4:1 180 degrees from 275 33
center at 290 33
\ellipticalarc axes ratio 4:1 -145 degrees from 280 30
center at 290 29
{\setquadratic 
\plot 150 40    163 38    175 40   /
\plot 150 20    163 22    175 20  /
\plot 235 40    248 38    260 40   /
\plot 235 20    248 22    260 20  /
}
\endpicture
}
\end{center}



\pagebreak 

 Einstein's gravitational field equations:
 
 

\bigskip
{\Large

$$
R_{\mu \nu} -\frac{R}{2}g_{\mu\nu}= {\frac{8\pi G}{c^4}} T_{\mu\nu}
$$
}


\pagebreak 

Hodge theorem:


\bigskip
{\Large

$$
H^p(M) = \{ \varphi \in \Omega^p(M)~|~ d\varphi = 0, ~d\star \varphi = 0\}
$$
}


\pagebreak 

Dirac Equation:


\bigskip
{\Large
\begin{eqnarray*}
\left(-i\hbar \gamma^\mu \nabla_\mu +mc\right)  \psi&=&0\\
\gamma^\mu\gamma^\nu+ \gamma^\nu\gamma^\mu&=&g^{\mu\nu}
\end{eqnarray*}

}


\pagebreak 

Seiberg-Witten Equations:


\bigskip
{\Large
\begin{eqnarray*}
\Dirac_A  \psi&=&0\\
F_A^+&=&-\frac{1}{2} \psi \odot \bar{\psi} 
\end{eqnarray*}

}

\pagebreak 


Elliptic curve, and uniformizing elliptic integral:

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center at 30 120 
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center at  33 120
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center at 30 68
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center at 30 68
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\plot   30 62  30 82  /
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}
\endpicture
}
\end{center}


{\Large
$$
F(z)=  \int_{z_0}^z \frac{d\zeta}{\sqrt{\zeta(\zeta -1)(\zeta -\lambda)}}
$$
}

\end{document}