Tractability of computation

• Creativity and Art

  1.

  Knowing when to pattern

  2.

  Symbol attachment and creation;
  patterns/symbols as revealers and
  concealers

  3.

  Levels of patterning

• Multiple patterns and selection
  (x - 1)(x - 2)(x - 3) - 6
  x³ - 6x² + 11x - 12
  (x - 4)(x² - 2x + 3)
• Adaptive pattern recognition
• Are the patterns really there?

Some history

• Physics
• Philosophy (theory of knowledge)
• Mathematics

  1.

  Matrix manipulation

  2.

  Topology

  3.

  Algebra

  4.

  Lie groups

  5.

  Manifolds and relativity theory

  6.

  Algebraic topology

We have the map $b_n: \Sigma^2U(n) \rightarrow SU(n+1)$
given by

$$b_n(g, r, s) = \left[ i(g), v_n(r, s) \right]$$

where i(g) is the inclusion, $\left[g, h\right] = ghg^{-1}h^{-1}$
and

v_n(r,s) =

$$\left[ \begin{array}{cccccc} \alpha & 0 & 0 & \cdots & 0 & \... ...line{\beta} & -(-\overline{\alpha})^n \\ \end{array} \right]$$

where

$$\alpha = \alpha(r,s) = \cos(\pi r) + i \sin(\pi r)\cos(\pi s)$$

$$\beta = \beta(r,s) = i \sin(\pi r)\sin(\pi s)$$

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

What's wrong in computing today

• Not enough resolution on displays
• Not enough processing power and memory
• Not enough parallelism
• Software tools are ``flat'' and sequential
  rather than hierarchical

The intelligent mathematical assistant

• Adaptive symbolic input and output
• Strong basic skills (all of arithmetic
  through college calculus and
  elementary discrete structures)
• First order logic capabilities
• Adaptive ``patterning'' and ``symboling''
• Elementary hypothesis generation
  and testing