I’ll tell the cameraman. I’m not sure the position of the camera was worse; I think it was similar to the first video. In the second video the camera was more centered, but further away, and I thought my voice was hard to hear. So, in the third video I tried to put the camera in the same position as for the first video. Maybe the cameraman was just getting too excited about panning and zooming.

]]>other than that, I really like the seminar, and i will keep watching it. thank you for filming it. ]]>

By the way, concerning the category of simplicial complexes: I’m currently studying The joy of cats (for unrelated reasons). They have some further results on , the category of simplicial complexes:

]]>28.21 Definition: A well-fibred topological construct for which is a quasitopos is called a

topological universe.[..]

28.23 Examples: [and some other cats] are topological universes.

I’m almost sure that you already know about it, but it reminds me this very cool paper http://link.springer.com/article/10.1007%2FBF01843493 by Goguen in automata theory. He proves that “blackboxing an automaton” is the left adjoint of minimal realization, i.e., the simplest automaton exhibiting this behaviour.

Would that make sense to do something similar here ? The right adjoint would be something that finds the simplest passive network matching the input/output data …

Anyway, thanks for these blog posts.

]]>I don’t see why orderings are needed for what we’re talking about here: networked dynamical systems.

What I am trying to say that in networks of Hamiltonian systems you would want to assign potentials to unordered collections of vertices. That means, as you said, that a network in this case is either a decorated symmetric complexes or a decorated symmetric set.

Now, here is an “obvious” interesting question: if these are the objects, what are the morphisms?

If your dynamical system is not Hamiltonian then you could have one part of the system drive another, and that looks directional to me. That’s why Golubitsky and Stewart and their collaborators have directed graphs… (OK, perhaps I should not put it quite so strongly, but it should be roughly correct)

]]>When one says “simplicial complex”, are the vertices necessarily ordered?

I don’t know. Some people say a simplicial complex is a simplicial set with an extra property; with this sort of definition, the vertices in any simplex are ordered. This is what you’d want in algebraic topology: you want a simplex to determine a chain, so it had better have an orientation.

But it seems standard to say a simplicial complex is a set equipped with an arbitrary collection of finite subsets. That’s quite pretty, and now the vertices in any simplex (i.e., the points in one of the finite subsets) are unordered. This is the sort of simplicial complex I look at in my paper with Alex Hoffnung.

What do you call an

unorderedsimplicial complex?

How about an “unordered simplicial complex”?

I don’t know all the standard terminology here. But an “unordered simplicial set” is called a **symmetric set**: it’s a contravariant functor from the category of finite sets to . So, if a simplicial complex is a simplicial set with a special property, the analogous thing for symmetric sets deserves to be called a **symmetric complex**.

I don’t see why orderings are needed for what we’re talking about here: networked dynamical systems. Is that part of what you’re saying?

]]>