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Scalable spaces
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\emph{Scalable spaces} are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality. They are also characterized by particularly nice behavior from the point of view of quantitative homotopy theory. Among other results, we show that spaces which are formal but not scalable provide counterexamples to Gromov's long-standing conjecture on distortion in higher homotopy groups.
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A hardness of approximation result in metric geometry
It is NP-hard to approximate the hyperspherical radius of triangulated surfaces and triangulated high-dimensional spheres to within any almost-polynomial factor.
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