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254A, Notes 9: Applications of the structural theory of approximate groups

In the last set of notes, we obtained the following structural theorem concerning approximate groups:

Theorem 1 Let be a finite -approximate group. Then there exists a coset nilprogression of rank and step contained in , such that is covered by left-translates of (and hence also by right-translates of ).

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285G, Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman reduced volume

We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known as the Perelman reduced volume. We saw in the previous lecture that the monotonicity for Perelman entropy was ultimately derived (after some twists and turns) from the monotonicity of a potential under gradient flow. 3,708 more words