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Abstract: We shall look at classes of structures that are invariant under some basic contructions. Among interesting situations are those when the classes are invariant under quotients, sums and products. The first two conditions mean that the class is so called coreflective, i.e., every space has a lower modification in the class. To find whether, or how much, such classes are productive need not be easy. We shall present some results showing limits of productivity in these situations in topological or uniform spaces, topological groups and locally convex spaces. Submeasurable cardinals play a role there. We can mention that it is still unknown whether there is a nontrivial class of topological spaces that is closed under quotients, disjoint sums and all products. |
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Abstract: Let F be a closed surface of genus \geq 2. Harvey defined the curve complex, a simplicial complex whose vertices are essential simple closed curves in F up to isotopy, and whose simplicies are represented by pairwise disjoint curves. Minsky and Masur showed that this complex is Gromov hyperbolic. I will present a recent simpler proof by Bowditch. |
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