I recently finished the first draft of the the first of my books, entitled “Hilbert’s fifth problem and related topics“, based on… 26 more words

## Tags » 254A - Hilbert's Fifth Problem

#### 254A, addendum: Some notes on nilprogressions

This is an addendum to last quarter’s course notes on Hilbert’s fifth problem, which I am in the process of reviewing in order to transcribe them into a book (as was done similarly for several other sets of lecture notes on this blog). 2,854 more words

#### 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:

3,423 more wordsTheorem 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 ).

#### 254A, Notes 8: The microstructure of approximate groups

A common theme in mathematical analysis (particularly in analysis of a “geometric” or “statistical” flavour) is the interplay between “macroscopic” and “microscopic” scales. These terms are somewhat vague and imprecise, and their interpretation depends on the context and also on one’s choice of normalisations, but if one uses a “macroscopic” normalisation, “macroscopic” scales correspond to scales that are comparable to unit size (i.e. 6,993 more words