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:

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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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


254A, Notes 7: Models of ultra approximate groups

In the previous set of notes, we introduced the notion of an ultra approximate group – an ultraproduct of finite -approximate groups for some independent of , where each -approximate group may lie in a distinct ambient group . 8,358 more words


254A, Notes 6: Ultraproducts as a bridge between hard analysis and soft analysis

Roughly speaking, mathematical analysis can be divided into two major styles, namely hard analysis and soft analysis. The precise distinction between the two types of analysis is imprecise (and in some cases one may use a blend the two styles), but some key differences can be listed as follows. 9,063 more words


254A, Notes 5: The structure of locally compact groups, and Hilbert's fifth problem

In the previous notes, we established the Gleason-Yamabe theorem:

Theorem 1 (Gleason-Yamabe theorem) Let be a locally compact group. Then, for any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that is isomorphic to a Lie group.

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