[Haskell-cafe] Re: [Haskell] small boys performance

[Dan Piponi]

On 3/14/07, Andrzej Jaworski <himself at poczta.nom.pl> wrote:
> 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