Updated: 2011-06-25 22:23:41
Over the past few months or so, I have been brushing up on my Lie group theory, as part of my project to fully understand the theory surrounding Hilbert’s fifth problem. Every so often, I encounter a basic fact in Lie theory which requires a slightly non-trivial “trick” to prove; I am recording two of [...]
Updated: 2011-06-21 23:42:24
Let be a Lie group with Lie algebra . As is well known, the exponential map is a local homeomorphism near the identity. As such, the group law on can be locally pulled back to an operation defined on a neighbourhood of the identity in , defined as where is the local inverse of the [...]
Updated: 2011-06-17 23:13:49
Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect [...]
Updated: 2011-06-13 21:03:47
We recall Brouwer’s famous fixed point theorem: Theorem 1 (Brouwer fixed point theorem) Let be a continuous function on the unit ball in a Euclidean space . Then has at least one fixed point, thus there exists with . This theorem has many proofs, most of which revolve (either explicitly or implicitly) around the notion [...]
Updated: 2011-06-07 16:31:22
In the last few months, I have been working my way through the theory behind the solution to Hilbert’s fifth problem, as I (together with Emmanuel Breuillard, Ben Green, and Tom Sanders) have found this theory to be useful in obtaining noncommutative inverse sumset theorems in arbitrary groups; I hope to be able to report [...]
Updated: 2011-06-04 04:53:34
In 1977, Furstenberg established his multiple recurrence theorem: Theorem 1 (Furstenberg multiple recurrence) Let be a measure-preserving system, thus is a probability space and is a measure-preserving bijection such that and are both measurable. Let be a measurable subset of of positive measure . Then for any , there exists such that Equivalently, there exists [...]
