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I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook): Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w …

Where can I find a reference for the following fact, or as close as possible to it? Let $G$ be a semisimple compact real Lie group with rank $r$, let $T$ be a maximal torus in $G$, let $\mathfrak{ …

According to [1] and [2], it is “well-known” that a compact Lie group $G$ has the same rational homology, and according to [2] is even rational homotopy equivalent, to the product $\mathbb{S}^{2m_1+1} …

Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me: as a quotient of a semisimple real Lie group $G$ …

Veblen $\phi$ functions extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't repea …

Let $\mathbf{C}$ be a closed symmetric monoidal category (I probably need even less than this; the examples I have in mind are simply the category of modules over a commutative ring and the category o …

Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \ma …

Is the following statement true, and if it is, does someone have a reference? Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, =Sto …

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try …

Stanley Hershel Stahl did his PhD thesis at the University of Michigan, under the direction of Peter Hinman, and wrote a dissertation in 1974 with title "Classes of Primitive Recursive Ordinal Functio …

When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ stru …

In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense. Now consider the following strengthening of the Baire p …

For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the prob …

Short question: Is there a standard term for a set $F$ such that there does not exist a surjection $F \twoheadrightarrow \omega$ (in the context of ZF)? More detailed version: Consider the following f …

I would like to learn about character theory of finite groups of Lie type and some Deligne-Lusztig theory. The classic textbook on the subject seems to be Roger W. Carter's Finite Groups of Lie Type: …