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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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Does the mean curvature flow naturally come with less applications than intrinsic curvature ...

Here is a list of some topological and geometric applications: Huisken-Ilmanen used inverse MCF to prove the Riemannian Penrose inequality: https://projecteuclid.org/euclid.jdg/1090349447 Huisken-Si …
Robert Haslhofer's user avatar