Abstract: This is the second part of our discussion of singularities of weighted projective spaces and their resolutions.

Abstract: We continue our exposition of the Schur-Horn-Kostant theorem for Symplectomorphisms groups of toric manifolds

Abstract: More than 50 years ago Michel Kervaire constructed a manifold with no differentiable structure. Constructing manifolds is the easy part - the trick is to construct an invariant, in this case the Arf-Kervaire invariant, which guarantees the existence or otherwise of the property one is after. The connection between framed manifolds and stable homotopy groups led (circa 1960) to the problem: ``Do there exist framed manifolds of odd Arf-Kervaire invariant?''. Recently Hill-Hopkins-Ravenel proved that the outcome is: ``mostly framed manifolds of Arf-Kervaire invariant one do not exist''.

Here is what Lewis Carroll has to say about non-existent things!

``I know what you're thinking about,'' said Tweedledum; ``but it isn't so, nohow.'' ``Contrariwise,'' continued Tweedledee,``if it was so, it might be; and if it were so, it would be; but as it isn't; it ain't. That's logic.''