%Slide 1
\begin{slide}{}
\begin{center}
{\large {\bf Intelligent Patterning }} \\
  or \\
{\large Why I've been doing computer science }
\end{center}
\end{slide}

%Slide 2
\begin{slide}{}

{\large Brief overview of where I'm headed: }

\begin{itemize}
	\item General problem solving
	\item Pattern recognition
	\item Symbols and signs
	\item Intelligent patterning
	\item Some history
	\item What's wrong in computing today
	\item The intelligent mathematical assistant
\end{itemize}

\end{slide}

%Slide 3
\begin{slide}{}

{\large General problem solving }

\begin{itemize}
	\item Understanding the problem
	\begin{enumerate}
		\item Problem context and statement of \newline
			the problem
		\item Solving the right problem (ill-posed and \newline
			ill-conditioned problems)
		\item Preconceptions
		\item Language and restating the problem
	\end{enumerate}
	\item The role of experience
	\begin{enumerate}
		\item Similar problems and analogy
		\item Appropriate tools
		\item Specific experience
	\end{enumerate}
\end{itemize}

\end{slide}

%Slide 4
\begin{slide}{}

\begin{itemize}
	\item Three basic methods
	\begin{enumerate}
		\item Plug and grind
		\item Guess and prove
		\item Look it up
	\end{enumerate}
	\item Hypothesis generation and testing
	\begin{enumerate}
		\item Flexibility and freedom --- willingness to
			try and fail
		\item Recognizing blind alleys, and the value of
			exploring
		\item Appropriate hypotheses
		\item Lateral thinking
	\end{enumerate}
\end{itemize}

\end{slide}

%Slide 5
\begin{slide}{}

\begin{itemize}
	\item Recognizing solutions
	\begin{enumerate}
		\item ``A'' solution vs. ``the'' solution
		\item Useful solutions
		\item When a ``solution'' solves an un-posed, but more
			significant problem
	\end{enumerate}
\end{itemize}

\end{slide}

%Slide 6
\begin{slide}{}

{\large Pattern recognition }

\begin{itemize}
	\item Images (``visual patterns'') vs. \newline
		``syntactic'' patterns
	\item Symbols as patterns, and symbols as \newline
		pattern labels
	\item Patterns of symbols
	\item Hierarchies of patterns, and symbols as \newline
		tools for recognizing patterns
	\item Pattern manipulation
	\item Learning to recognize patterns, and pattern \newline
		recognition as learning
\end{itemize}

\end{slide}

%Slide 7
\begin{slide}{}

{\large Pattern recognition examples }

\begin{itemize}
	\item What number comes next in the sequence? \newline
		1, 1, 2, 3, 5, 8, 13, \ldots
	\item What number comes next in the sequence? \newline
		8, 5, 4, 9, 1, 7, 6, 3, \ldots
	\item What letter comes next in the sequence? \newline
		E, T, A, O, I, N, S, H, \ldots
	\item In which row does Z go? \newline
		A, E, F, H, I, K, L, M, N, T, V, W, X, Y \newline
		B, C, D, G, J, O, P, Q, R, S, U
	\item What letter comes next in the sequence? \newline
		W, L, C, N, I, T, \ldots
\end{itemize}
\end{slide}

%Slide 8
\begin{slide}{}

{\large Symbols and signs }

\begin{itemize}
	\item The utility and power of symbols
	\item Choosing symbols, naming and pointing
	\item Symbols as ``chunking'' tools
	\item When to use symbols
	\begin{enumerate}
		\item The importance of anonymity (e.g., the \newline
			lambda calculus)
		\item Place holders (variables)
		\item Temporary and tentative symbols
	\end{enumerate}
	\item Signs, symbols, content and meaning
\end{itemize}

\end{slide}

%Slide 9
\begin{slide}{}

{\large Intelligent patterning }

\begin{itemize}
	\item Creativity and Art
	\begin{enumerate}
		\item Knowing when to pattern
		\item Symbol attachment and creation; \newline 
			patterns/symbols as revealers and \newline
			concealers
		\item Levels of patterning
	\end{enumerate}
	\item Multiple patterns and selection \newline
		$ (x - 1)(x - 2)(x - 3) - 6 $ \newline
		$ x^3 - 6x^2 + 11x - 12 $ \newline
		$ (x - 4)(x^2 - 2x + 3) $
	\item Adaptive pattern recognition
	\item Are the patterns really there?
\end{itemize}

\end{slide}

%Slide 10
\begin{slide}{}

{\large Some history }

\begin{itemize}
	\item Physics
	\item Philosophy (theory of knowledge)
	\item Mathematics
	\begin{enumerate}
		\item Matrix manipulation
		\item Topology
		\item Algebra
		\item Lie groups
		\item Manifolds and relativity theory
		\item Algebraic topology
	\end{enumerate}
\end{itemize}

\end{slide}

%Slide 11
\begin{slide}{}

We have the map
	$ b_n: \Sigma^2U(n) \rightarrow SU(n+1) $ \newline
given by
	\[ b_n(g, r, s) = \left[ i(g), v_n(r, s) \right] \]
where $i(g)$ is the inclusion,
	$\left[g, h\right] = ghg^{-1}h^{-1}$ \newline
and

$ v_n(r,s) = $

{\small
\[
\left[ \begin{array}{cccccc}
	\alpha & 0 & 0 & \cdots & 0 & \beta (-\overline{\alpha})^0 \\
	\beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha  &  0  & \cdots & 0 &
	   \beta (-\overline{\alpha})^1 \\
	\beta (-\overline{\alpha})^1\overline{\beta} &
	   \beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha & \cdots & 0 & \beta (-\overline{\alpha})^2 \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\beta (-\overline{\alpha})^{n-1}\overline{\beta} &
	   \beta (-\overline{\alpha})^{n-2}\overline{\beta} &
	   \cdots &  \cdots & \alpha &
	   \beta (-\overline{\alpha})^n \\
	-(-\overline{\alpha})^n\overline{\beta} &
	   -(-\overline{\alpha})^{n-1}\overline{\beta} &
	   \cdots &  \cdots & -(-\overline{\alpha})^0 
	   \overline{\beta} & -(-\overline{\alpha})^n \\
	\end{array} \right]
\]
}

{\small
where
\[ \alpha = \alpha(r,s) =
	\cos(\pi r) + i \sin(\pi r)\cos(\pi s) \]
\[ \beta = \beta(r,s) = i \sin(\pi r)\sin(\pi s) \]
}

\end{slide}

%Slide 12
\begin{slide}{}
{\tiny
\begin{verbatim}
We have the map
	$ b_n: \Sigma^2U(n) \rightarrow SU(n+1) $ \newline
given by
	\[ b_n(g, r, s) = \left[ i(g), v_n(r, s) \right] \]
where $i(g)$ is the inclusion,
	$\left[g, h\right] = ghg^{-1}h^{-1}$ \newline
and

$ v_n(r,s) = $

\[
\left[ \begin{array}{cccccc}
	\alpha & 0 & 0 & \cdots & 0 & \beta (-\overline{\alpha})^0 \\
	\beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha  &  0  & \cdots & 0 &
	   \beta (-\overline{\alpha})^1 \\
	\beta (-\overline{\alpha})^1\overline{\beta} &
	   \beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha & \cdots & 0 & \beta (-\overline{\alpha})^2 \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\beta (-\overline{\alpha})^{n-1}\overline{\beta} &
	   \beta (-\overline{\alpha})^{n-2}\overline{\beta} &
	   \cdots &  \cdots & \alpha &
	   \beta (-\overline{\alpha})^n \\
	-(-\overline{\alpha})^n\overline{\beta} &
	   -(-\overline{\alpha})^{n-1}\overline{\beta} &
	   \cdots &  \cdots & -(-\overline{\alpha})^0 
	   \overline{\beta} & -(-\overline{\alpha})^n \\
	\end{array} \right]
\]
where
\[ \alpha = \alpha(r,s) =
	\cos(\pi r) + i \sin(\pi r)\cos(\pi s) \]
\[ \beta = \beta(r,s) = i \sin(\pi r)\sin(\pi s) \]

\end{verbatim}
}
\end{slide}

%Slide 14
\begin{slide}{}

{\large What's wrong in computing today }

\begin{itemize}
	\item Not enough resolution on displays
	\item Not enough processing power and memory
	\item Not enough parallelism
	\item Software tools are ``flat'' and sequential \newline
		rather than hierarchical
\end{itemize}

\end{slide}

%Slide 12
\begin{slide}{}

{\large The intelligent mathematical assistant }

\begin{itemize}
	\item Adaptive symbolic input and output
	\item Strong basic skills (all of arithmetic \newline
		through college calculus and \newline
		elementary discrete structures)
	\item First order logic capabilities
	\item Adaptive ``patterning'' and ``symboling''
	\item Elementary hypothesis generation \newline
		and testing
\end{itemize}
\end{slide}
%