Comments on: Kock on synthetic differential geometry

By: Ars Mathematica » Blog Archive »

Ars Mathematica » Blog Archive » — Thu, 03 Aug 2006 05:08:32 +0000

[…] If anyone is interested in some more synthetic differential geometric goodness, the point of view of the book Natural Operations in Differential Geometry by Ivan Kolar, Jan Slovak and Peter W. Michor, while couched in a more traditional language, is quite close to that of synthetic differential geometry. In Natural Operations, the authors are trying to classify functors on the category of differentiable manifolds. Synthetic differential geometry tries to define a larger category so that those functors become representable. […]

By: sigfpe

sigfpe — Thu, 27 Jul 2006 21:36:17 +0000

The method lends itself to an elegant approach to computing derivatives numerically. I write a little about it here. You end up with surprisingly simple code. As far as I can see, almost everyone who needs to compute derivatives of complicated functions in software ends up using some form of finite differences or else working with gigantic and inefficient expressions generated by Mathematica or Maple. But computers are just as happy to work over R[e]/(e^2) as over R and the latter approach works very well in practice. One thing I haven't seen in publications anywhere is that they also give a nice way to work, computationally, with tangents spaces of varieties. For example, if you've written software to multiply elements of a Lie group (eg. rotations represented as 3x3 matrics), and then reimplement the same code to work over R[e]/(e^2) instead of R, then code to compute the Lie bracket "drops out" for free. It gives a nice way to unify and simplify some geometric algorithms.

By: Walt

Walt — Thu, 27 Jul 2006 03:53:22 +0000

The flavor is much closer to that of algebraic geometry. At the root of their construction, they define a category of rings of differentiable functions, and take the opposite category. So for example, they define a ring with a nilpotent element by taking the ring of differential functions from R to R, and modding out by all functions whose derivatives are zero at zero. (This just turns out to be the ring R[e], where e^2 = 0.)

By: michael

michael — Wed, 26 Jul 2006 18:25:54 +0000

I suppose I should just read the book, but how are the infintesimals contructed? An Ultrapower contruction or something a little less concrete?

By: PeterMcB

PeterMcB — Wed, 26 Jul 2006 17:01:24 +0000

I first came across Kock’s book several years ago. It is a very nice treatment.