I am looking for a good reference for fibre bundles on differential manifolds, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of Ehresmann connections). Could anyone advise me on this? I have looked at the book Fibre Bundles by Hausmöller, but it isn’t quite what I want. Thanks in advance.

I ran into the following description of Galois theory in Gelfand and Manin’s Methods of Homological Algebra. But this looks like nothing like the Galois theory I know that deals with subgroups and field extensions. I wonder if there is any great self contained books that treats this topic thoroughly enough so that I can […]

The Urysohn’s metrization theorem states that for any second countable and regular $(X,\tau)$ there exists a topological embedding $f:X\to [0,1]^{\mathbb{N}}$. What puzzles me is not the theorem itself, but why (and if) $f$ can be chosen so that the image $f(X)$ is a Borel subset of $[0,1]^{\mathbb{N}}$. This is something that is not clear to […]

I am currently working on something related to the character theory of the group of unipotent upper triangular matrices with elements in a finite field. I have seen in many papers on the topic the statement that determining the irreducible characters and conjugacy classes of these groups is a “wild” problem, but never with a […]

I would like references for algorithms solving the conjugacy problem in $F_n$ (the free group on $n$ generators)?

My ultimate goal is to see how the point of infinity and an arbitrary transform in Riemann sphere can lead to what consequences in dynamical systems, and it seems that harmonic analysis plays a crucial role in between since it connects Fourier transform and spherical harmonics, Hilbert space and functional analysis, topology, group, graph and […]

My understanding is that Solovay (1970)’s relative consistency shows that if ZFC+I has a model then ZF+DC has a model in which every subset of the reals is Lebesgue measurable (and hence $\sigma$-additive). I was wondering if there a relative consistency result that shows that ZF + {some weaker than AC/DC condition} proves the existence […]

This is an question from the book “A concise introduction to Pure Mathematics”. I understand that it looks like a homework question but it’s the first chapter and there are no answers for even questions. As I am independently trying to make my way through a bit of maths I was hoping I could get […]

Let say from the beginning that my background is category of modules over a ring. So I know that if we take a given (nice) scheme $X$, then category of sheaves on $X$ is Grothendieck, so it must have enough injective, but it may fail to have projective. The question is then, when it is […]

I have been able to find these two but I don’t know how valuable they are as a reference, “Problems and examples in differential equations” By Biler, and, “Partial Differential Equations through Examples and Exercises” by Pap. However, I don’t know if these are any good, or the ones that I’m looking for, maybe you […]

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