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\def\DDirac{\not\hspace{-1.50mm}D}

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\begin{document}


\noindent{N. \it Atiyah-Singer Theorem for Twisted Dirac Operator:} 


$$\dim \ker \slash \! \! \! \! {D}_E - \dim \mbox{coker}~  
\slash \! \! \! \! {D}_E = \int_M \hat{A}(M) \cdot ch (E)$$
\vs

\noindent{M. \it Dirac Equation:}


$$
\left(i\DDirac - m\right)  \psi=0
$$
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\noindent{39. \it Cauchy's Integral Formula:}

$$f(z) = \frac{\ds 1}{\ds 2\pi i}\oint_C \frac{\ds f(\zeta)}{\ds \zeta - z}~d\zeta$$
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\newpage

\noindent{41. \it Graphic with Feynman diagrams and surfaces}
[insert graphic]
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\noindent{42? \it Navier-Stokes equation:}
$$  \p_t v_i + v_j \p_j v_i = -\p_i p + \nu \p_j \p_j v_i  $$
\vs



\noindent{43. \it Kolmogorov Law:}                  
$$   E(k) \sim \varepsilon^{2/3} k^{-5/3} $$
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\noindent{47. \it The entropy formula:}
$$ S =  -\sum_i p_i \log p_i  $$ 
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\noindent{49. \it Bott Periodicity:}

$$ \pi_i({\bf {\rm U}}) = \pi_{i+2}({\bf {\rm U}})$$
$$ \pi_i({\bf {\rm O}}) = \pi_{i+8}({\bf {\rm O}})$$
$$ {\mathbb Z}_2, {\mathbb Z}_2, 0, {\mathbb Z}, 0, 0, 0, {\mathbb Z} $$


\noindent{50. \it Ricci flow:}
$$\frac{\partial g_{t}}{\partial t} = -2Ric(g_{t})$$ 
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\noindent{51. \it Apollonian fractal:}
\begin{figure}[htp]
\centering \includegraphics[width=1in]{scott_apollonian.eps}
\end{figure}
\newpage

\noindent{53? \it Pascal's triangle, Fibonacci numbers:}
\begin{figure}[htp]
\centering \includegraphics[width=1in]{triangle.eps}
\end{figure}
\vs





\noindent{55. \it Aharonov-Bohm Effect:}

\vs

$$\int_{C_2} {\vec A}\cdot d{\vec\ell} - \int_{C_1} {\vec A}\cdot d{\vec\ell}  
= \frac{1}{2\pi}\Phi$$

[insert figure]
\vs


\noindent{56. \it Group law on cubic:}

\begin{figure}[htp]
\centering \includegraphics[width=1in]{cubic.eps}
\end{figure}
$${\bf\tt a} + {\bf\tt b} + {\bf\tt c} = 0$$

\noindent{57? \it Riemann-Roch-Hirzebruch:}
$$
\sum_{k=0}^n (-1)^k{\rm dim} \, H^k(X,E)\ \ =\ \ \int_X ch\,E \cup Todd(X)
$$
\newpage

\noindent{58. \it Platonic solids, Euler characteristic:}
\begin{figure}[htp]
\centering \includegraphics[width=1in]{tetrahedron.eps}
\centering \includegraphics[width=1in]{cube.eps}
\centering \includegraphics[width=1in]{octahedron.eps}
\centering \includegraphics[width=1in]{dodecahedron.eps}
\centering \includegraphics[width=1in]{icosahedron.eps}
\end{figure}
$$ V - E + F = 2$$
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\noindent{61? \it Mandelbrot Set:}
\begin{figure}[htp]
\centering \includegraphics[width=1in]{mandelbrot.eps}
\end{figure}
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\noindent{63. \it Eratosthenes' measurement of radius of Earth:}
\begin{figure}[htp]
\centering \includegraphics[width=1in]{milnor-eratos.ps}
\end{figure}
\vs

\noindent{64? \it Prime number Theorem:}
$$ \pi(x)~\sim~\frac{x}{\ln x}$$
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\noindent{65. \it Supergravity:}
$${\mathcal L}=R-\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}D_{\rho}\psi_{\sigma}$$
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\newpage
\noindent{66. \it Fourier transform:}


$$\hat{f}(\xi) = \int f(x) e^{2\pi i x\cdot\xi} dx \hspace{1in} f(x) = \int \hat{f}(\xi) e^{-2\pi i x\cdot\xi} d\xi$$ 
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\noindent {67. \it Euler's summation for $\zeta(2)$:}

$$ 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\ds \pi^2}{\ds 6}$$
\vs



\noindent{68. \it Kepler's laws:}

\begin{figure}[htp]
\centering \includegraphics[width=1in]{milnor-keppic2.ps}
\end{figure}
%\centerline{Kepler's Laws:\qquad $r\;d\theta/dt={\rm constant}\,,
%\qquad \oint dt \sim s^{3/2}$}
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\noindent{70. \it dd}
$$\partial\circ\partial = 0$$
[insert graphic? exploded tetrahedron?] 

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\noindent{71. \it Poincar\'{e}}
$$ \pi_1M^3 = 0 \hspace{.1in} \Rightarrow \hspace{.1in} M^3 \approx S^3 $$
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\noindent{72. \it Fermat}
$$ x^n + y^n \neq z^n $$
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\end{document}

