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