1. First conjectured certain conformal content in SOM and prove this content in 2000.
{Kohonen’s book 1995, mentioned quasiconformal mapping without any references. But How?}
Then we proposed,
use conformal mapping (angle invariance)
to preserve
geometrical topological relations in SOM.
This may (possible) lead to settle
differentiation foundations for SOM.
Local energy in each simplex is possible. Laplace equation for conformal mapping in each simplex.

2. resolve local difficult competition.
Conformal SOM can
resolve local difficult competition.
preserve geometrical angle relations.
keep integrity of local topological relations.
..This is consistent with the idea of SOM.
provide
a continuous and differentiable surface for SOM
tracing continuous state trajectory possible,
dynamic trajectory patterns
(texture) predictions * (marble lines, wood grains)
other than clustering, Gaussian mixture, CWM,…
nonlinear and flexible data reduction *
other than spline, curve fitting,…
divide and conquer *
for both global and “local” representations.

6. Demo
Economic state indicator on neuronic map, 9’th International Conference on Neural Information Processing,
ICONIP, vol.2, pp ,
Nov. 18~22, 2002, Singapore

12. Why do we prefer CSM? A simple flower vessel or rabbit ear or
differentiable surface or genus zero surface
can crush methods:
LLE, ISOMAP, GTM, C-ISOMAP,
MDS, …
tracking continuous states
resolving various difficult competitions