I have never had an experience with creating simulations or 3D objects, but I want to start learning them, and create a small application, which will simulate topological objects in 3D. What I mean by “topological object”, is mathematical topology (algebraic/general topology), meaning torus, knot, Möbius strip, etc. So, I don’t mean anything like network topology.
I have been searching over the Internet for some example codes regarding them, but I wasn’t able to find anything useful. If you could offer me some materials, I would be glad. On the other hand, I want to hear your opinion about which programming language/paradigm/extension should I use? I also plan to use CUDA with the project for achieving speed-ups.
I don’t think CUDA is really much use for computional algebraic/general topology. Sure, you can use it to mess around with homology groups etc., but that’s rather algebra than topology, which itself tends to be too abstract/”dynamic” to really benefit from SIMD. If you don’t have a clear idea, I’d first try some CPU implementations and only port to CUDA as a later optimisation.
Anyway, what you describe sounds rather like you’re mainly interested in creating visual representations of topological spaces, i.e. in giving concrete embeddings T → ℝ³. That’s rather in the realm of differential topology, which I reckon could quite well make use of gpgpu processing. However for the final “visualisation step”, you want to use something more specific; openGL + GLUT is fine. You can use that from many languages, I’d recommend Haskell (undisputedly great for everything mathematical), but C or C++ is of course closer to the library, you will find more examples and can easier get CUDA in.