On 3/14/07, Andrzej Jaworski
I am glad you are interested Dan. ... I do not intend to bore anybody with differential geometry but as I was pushed that far let me add that if Haskell was made to handle Riemannian geometry it could be useful in next generation machine learning research where logic, probability and geometry meet.
I believe that you can probably handle (pseudo-)Riemannian geometry in the framework sketched here: http://sigfpe.blogspot.com/2006/09/practical-synthetic-differential.html That only goes as far as playing with vector fields and Lie derivatives but I think that forms and tensors should fit just fine into that framework. There's a simple way to use types to represent tensor products, and that's sketched here: http://sigfpe.blogspot.com/2006/08/geometric-algebra-for-free_30.html (Forget that that's about geometric algebra, the thing I'm interested in is the tensor products.) So I'm guessing there's a way of combining these to give a framework for (pseudo-)Riemannian geometry. But it'd only be a good framework for answering certain types of questions - in particular for things like numerical simulation. The important thing is that you'd be able to read off accurate numerical values of quantities like curvatures without any need for symbolic algebra. -- Dan