Articles of reference request

Reference for principal bundles and related concepts

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 Hausm√∂ller, but it isn’t quite what I want. Thanks in advance.

Galois theory reference request

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 […]

Urysohn's metrization theorem and Borel image

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 […]

Reference for a “wild” problem

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 […]

The conjugacy problem of finitely generated free group

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

Roadmap to understand the link between harmonic analysis and Riemann sphere?

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 […]

ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$

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 […]

Prove by contradiction that a real number that is less than every positive real number cannot be posisitve

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 […]

When is the category of (quasi-coherent) sheaves of finite homological dimension?

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 […]

Supplemental reference request-Graduate level PDE problems and solutions book

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 […]