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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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The geometric explanation of **isotropic position**
A convex body $K$ in $\mathbb{R}^n$ is in isotropic position if, for all vectors $x \in \mathbb{R}^n$, we have
$$\frac{1}{\mathrm{vol}(K)}\int_K \langle x, y \rangle^2 dy = \|x\|^2.$$
My question: T …