A combinatorial graph contains much
information, its Laplace spectrum, its Cheeger constant,....
Other information can determine ways by which it can evolve, say
involving the parity of degrees and distance between its vertices; see
my article in Mind and Matter 2013. Okay, so I made up these
rules: it's not going to evolve under its own steam. But
wait a minute. Our brains do a pretty good job of functioning
autonomously. Most of the time we are reacting to external
stimuli, but when we sleep neurons are firing, electrical and chemical
signals are transmitted, a world is being created inside our
heads.
The human brain has been described as the most complex object in the
known universe, so to get a hold on how it functions, let's start with
something smaller: a combinatorial graph, or better still a quantum
network and try to undestand how form and change (a causal sequence)
can emerge from such structure. If we draw line segments on a
piece of paper in the right way, we see a cube, in fact two possible
cubes and we can flip from one to the other. The
visualization of a cube is known to psychologists as the
gestalt effect,
whereby we see whole forms rather than just a collection of line
segements. In my article Journal of Geometry and Physics 2013, I
formalize this effect with the notion of
geometric state. A graph has a different spectrum to its Laplace spectrum, which I call its
geometric spectrum,
which yields local realizations of the graph as an invariant
framework in Euclidean space (see Linear Algebra and its
Applications 2014). Projections of these realization to the
complex plane satisfy a quadratic difference equation, independently of
any similarity transformation (the spectrum is a parameter of this
equation). This removes the dependence on the ambient space and
reveals geometry, or form, as an emergent concept.
Introduce choice into this model and one has an example of quantum
interaction, an emerging discipline which applies the methods of
quantum theory to other areas of informatics
Quantum Interaction
. In the geometric model discussed above, quantum interaction
occurs in the two possible perceptions of the cube, in the quadratic
difference equation (whose linearization is like the Schroedinger
equation) satisfied by its projection to the complex plane, and the
fact that only these realizations solve this equation for the given
value of the geometric spectrum.
Other aspects of this project concern the reconstruction of 3D-images
and the efficient encoding of geometric information.