**Challenging computerizable problems:**

From stability of markets to supergravity

Cohomology, in particular Lie algebra cohomology, though a notion
understandable to any first year student of mathematics, is
hardly known to the general public. This is understandable: until computers
became sufficiently powerful to tackle such problems, the cohomological approach
to various vital problems of engineering (e.g., criteria of formal
solvability of differential equations) was hardly of any practical value,
even as a reason for a grant application.

Lately this situation has changed favourably, and previously wild problems can be
solved by means of computers (even based on MATHEMATICA, notoriously slow
and otherwise handicapped).

Recently I have realized how to formulate in terms of Lie algebra cohomology
the analog of the Riemann tensor for non-holonomic manifolds. Recall that
the term *nonholonomic* was coined by Hertz for dynamical systems with
nonintegrable constraints on velocities (like a bike or car: the point of
tangency of the weel with asphalt has velocity 0; the car with cruise
control ON gives an example of a non-linear constraint).

Apart from mechanical examples or examples given by electro-mechanical
devices, nonholonomic constraints manifest themselves in such practical examples
as the Cat's Problem (why the cat, being dropped, lands on its paws) or in
parking a 4-weeled vehicle.

Supergravity (whatever it is)
is manifestly nonholonomic. So in order to write SUGRA (the
analog of Einstein-Hilbert's equations), we have to know the nonholonomic
analog of the Riemann tensor (the left-hand side of the SUGRA eqs).

A still more interesting application of this cohomology --- of universal
interest I'd say --- is to stability of market economy. The point is: any
market is, as one can show, a non-holonomic system, whereas the sign of the
curvature tensor governs stability.

No preliminary knowledge is required. Of interest to majors in computer
science, differential geometry, economics.

*Dmitri Leites*