Thank you, much appreciated.

]]>Okay, I’ve added a video of the talk to this post!

]]>Yes, and I should get the video and put it on YouTube at some point. I’ll announce that on this page when it happens.

]]>It is indeed perfectly possible to have a disconnected symmetric space, but the usual definition requires connectedness so I didn’t want to mess with that in this talk. A really nice abstract definition of symmetric space would be ‘involutory quandle object in the category of smooth manifolds’, but then to recover Cartan’s classification we need to impose conditions like compactness, preservation of a Riemannian metric by the quandle operation a condition saying that each point is an isolated fixed point of and even more (some ‘irreducibility’ and simply-connectedness properties I didn’t want to talk about).

Anyway, all your guesses are right. There are three two-parameter families of compact symmetric spaces, the Grassmannians, ultimately because three Morita equivalence classes of Clifford algebras are not Morita equivalent to division algebras (whose representations are classified by a single natural number). Let me talk about the real case for simplicity. If you take the manifold of *all* subspaces of , it is a disjoint union of Grassmannians, but the whole thing is a quandle object, because it makes sense to reflect *any* subspace of across any subspace by applying the transformation that preserves and multiplies vectors orthogonal to by -1. So, it’s very nice—it just happens to be disconnected, as you noted.

Clearly for the purposes of classification, getting the connected ones is enough. And if I’m not mistaken, each connected component of one of these “disconnected compact symmetric spaces” should be a symmetric space, too.

In the slides, the example where the essential fibre is a disjoint union is where one has a two-parameter family, with the second parameter can be taken to have a finite range. I presume this is true for the other cases where one has these two parameter families (namely the real and complex Grassmannians)? One might make an argument that among the ten infinite families, they really should only have one parameter, just that three of those families consist of disconnected symmetric spaces (obviously with finitely many components).

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