\documentclass[12pt]{article} \newcommand{\vs}{\vspace{.1in}} \newcommand{\ds}{\displaystyle} \newcommand{\p}{\partial} \newcommand{\Tr}{\rm Tr} \usepackage[T1]{fontenc} \usepackage{yfonts} \usepackage{graphicx} \usepackage{latexsym} \usepackage{amsmath} \usepackage{amssymb} \def\Dirac{\not\hspace{-1.50mm}\partial} \def\DDirac{\not\hspace{-1.50mm}D} \usepackage{amsfonts} \usepackage{rawfonts} \usepackage{amsxtra} \usepackage{amscd} \usepackage{amsthm} \usepackage{eucal} %\usepackage{tikz} %\usetikzlibrary{arrows,decorations.pathmorphing,decorations.markings,trees,backgrounds,fit,calc,through} \usepackage{slashed} \input{prepictex} \input{pictex} \input{postpictex} \begin{document} \noindent{1. \it Primes and the zeta function:} $$ \prod_p \frac{1}{1-\frac{1}{p^s}} = \sum_{n=1}^{\infty}\frac{1}{n^s}$$ \vs \noindent{3. \it Yang-Baxter Equation} $$R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23}$$ \begin{figure}[htp] \centering \includegraphics[width=1in]{braid.eps} \end{figure} \vs \noindent{7. \it Newton's Law of Gravitation:} $$ F = \frac{\ds G m_1m_2}{\ds r^2}$$ \vs \noindent{8. \it Pythagoras' Theorem with no-word proof:} $$c^2 = a^2 + b^2 $$ \begin{figure}[ht] \centering \includegraphics[width=1in]{pythagoras.eps} \end{figure} \vs \noindent{10? \it Euler's Equation:} $$ e^{\ds i\pi} + 1 = 0$$ \vs \noindent{11? \it Schr\"{o}dinger's Equation:} $$ i\hbar \frac{\ds \partial\psi}{\ds \partial t} = -\frac{\ds \hbar^2}{\ds 2m} \nabla^2\psi + V\psi$$ \vs \newpage \noindent{12. \it Maxwell's Equations in Vacuum:} $$\begin{array}{l} \nabla \cdot {\bf B} = 0 \\ \nabla \times {\bf B} = \frac{\ds 1}{\ds c}\frac {\ds \partial {\bf E}} {\ds \partial t}\\ \end{array}~~~~~ \begin{array}{l} \nabla \cdot {\bf E} = 0 \\ \nabla \times {\bf E} = -\frac{\ds 1}{\ds c}\frac{\ds \partial {\bf B}} {\ds \partial t}\\ \end{array}$$ \vs \noindent{14. \it Einstein's General Relativity Equation: } $$R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu} = 8 \pi T_{\mu\nu}$$ \vs \noindent{16. \it Heisenberg Uncertainty Principle:} $$\Delta x \Delta p \geq \frac{\ds \hbar}{\ds 2}$$ \vs \noindent{17. \it Dynkin diagram of $E_8$:} \begin{figure}[htp] \centering \includegraphics[width=1in]{dynkin.eps} \end{figure} \vs \noindent{22. \it Stokes' Theorem} $$\int_{M}d\omega=\int_{\partial M}\omega$$ \noindent{24. \it Atiyah-Singer Theorem for Twisted Dirac Operator:} %$$\Dirac : \Gamma ({\mathbb S}_+\otimes E) \to \Gamma ({\mathbb S}_-\otimes E)$$ $$\dim \ker \slash \! \! \! \! {D}_E - \dim \mbox{coker}~ \slash \! \! \! \! {D}_E = \int_{M^{4k}} \hat{A}(M) \cdot ch (E)$$ \vs \noindent{26. \it Classical Gauss-Bonnet Theorem:} $$ \chi (M^{2})= \frac{1}{2\pi} \int_{M}K ~dA $$ \vs \noindent{27. \it Chern-Simons Action:} $$ S_{CS}=\frac{k}{2\pi}\int_M{\rm Tr}\left(\frac{1}{2}A\wedge d A+\frac{1}{3}A\wedge A\wedge A\right)$$ \vs \newpage \noindent{29. \it Snell's (Pascal's) Law:} \begin{figure}[htp] \centering \includegraphics[width=1in]{snell.eps} \end{figure} $$\sin i = n \sin r$$ \vs \noindent{32. \it Virasoro algebra:} $$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n} $$ \vs \noindent{34. \it Yang-Mills equations:} $$F = dA + A \wedge A$$ $$\nabla \wedge \star F =0$$ \vs \noindent{37. \it Dirac Equation:} $$ \begin{array}{rcl} \left(-i\hbar \gamma^\mu \nabla_\mu +mc\right) \psi&=&0\\ \gamma^\mu\gamma^\nu+ \gamma^\nu\gamma^\mu&=&2g^{\mu\nu} \end{array} $$ \vs $$ \left(i\Dirac - m\right) \psi=0 $$ \vs $$ \left(i\DDirac - m\right) \psi=0 $$ \vs \noindent{39. \it Cauchy's Integral Formula:} $$f(z) = \frac{\ds 1}{\ds 2\pi i}\oint_C \frac{\ds f(\zeta)}{\ds \zeta - z}~d\zeta$$ \vs \newpage \noindent{41. \it Graphic with Feynman diagrams and surfaces} [insert graphic] \vs \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:} $$ E(k) \sim \varepsilon^{2/3} k^{-5/3} $$ \vs \noindent{47. \it The entropy formula:} $$ S = -\sum_i p_i \log p_i $$ \vs \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:} $$\frac{\partial g_{t}}{\partial t} = -2Ric(g_{t})$$ \vs \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} $$ V - E + F = 2$$ \vs \noindent{61? \it Mandelbrot Set:} \begin{figure}[htp] \centering \includegraphics[width=1in]{mandelbrot.eps} \end{figure} \vs \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:} $$ \pi(x)~\sim~\frac{x}{\ln x}$$ \vs \noindent{65. \it Supergravity:} $${\mathcal L}=R-\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}D_{\rho}\psi_{\sigma}$$ \vs \newpage \noindent{66. \it Fourier transform:} $$\hat{f}(\xi) = \int f(x) e^{2\pi i x\cdot\xi} dx \hspace{1in} f(x) = \int \hat{f}(\xi) e^{-2\pi i x\cdot\xi} d\xi$$ \vs \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 $r\;d\theta/dt={\rm constant}\,, %\qquad \oint dt \sim s^{3/2}$} \vs \noindent{70. \it dd} $$\partial\circ\partial = 0$$ [insert graphic? exploded tetrahedron?] \vs \noindent{71. \it Poincar\'{e}} $$ \pi_1M^3 = 0 \hspace{.1in} \Rightarrow \hspace{.1in} M^3 \approx S^3 $$ \vs \noindent{72. \it Fermat} $$ x^n + y^n \neq z^n $$ \vs \end{document}