Friday, January 27, 2006

De Rham Cohomology and Computer Algebra

Computers are now good at a number of branches of mathematics. They've been performing symbolic integration, differentiation, summation and differential equation solving for decades. In recent years, with the advent of Grobner basis methods, they've become pretty competent at algebraic geometry. But one area has stood out in my mind as being unsuitable for computational methods and that's computing de Rham cohomology. In order to compute this we have to study spaces of differential forms on manifolds and these are large spaces. Given lots of pieces of differential form you can glue them to other bits of differential form using a partition of unity so in a sense you have a great freedom to mix and match forms. This is in stark contrast to algebraic geometry where spaces of functions frequently tend to be finite dimensional.

So I was surprised to find that in recent years computers have been calculating de Rham cohomology, and doing so using methods from algebraic geometry. In fact, functions to compute the de Rham cohomology of the complement of various algebraic varieties have been available in Macaulay2 for a while. There's a chapter of the Macaulay2 book available on the subject. The methods used are based on D-modules and discussed in the book Grobner Deformations of Hypergeometric Differential Equations.

I have no idea where the connection with hypergeometric series is coming from but it's interesting to see the connection with algebraic geometry. D-modules are rings with elements xi and di sith the proprty that xixj=xjxi and didj=djdi but that dixj-xjdiij. We can form ideals of these things and study them using Grobner basis methods. Given a smooth function we can form the ideal of elements of a D-module that annihilate the function by considering the di to be the differential operators d/dxi. This gives a connection between the methods of algebraic geometry and differential geometry. But that's as far as my knowledge extends for now.

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