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There is no set of all [compact] topological spaces. Given a set of topological spaces, consider its power set. This power set with the indiscrete topology is a compact topological space missing from …

The quote about how Grothendieck solved mathematical problems—emphasizing his approach of finding a standpoint where there are no special cases—is quite general. Without knowing enough about your expe …

A $\mathcal U(n)$-equivariant map of (finite-dimensional) representations $X \to Y$ is the same as a $\mathcal U(n)$-invariant of $X^* \otimes Y$, or equivalently a $\mathcal U(n)$-invariant function …

Assuming the axiom of choice, there is a (may I say very natural) uncountable set of real numbers that is measure-zero with regard to any $\sigma$-finite, complete, regular measure that measures all t …

There is no such function $h$, not just for the Riemann zeta function in the critical strip but for any non-constant holomorphic function $v:D\to\mathbb{C}$ defined on an open disk $D\subset\mathbb{C} …

This question is seven years old, but since I wanted to ask the same question and provide those examples for induction on prime numbers, I found this question and saw that the first example is missing …

Here is my take. Unlike Andy, I would not structure such a course around big theorems. In part, this is because your students simply do not have enough background to handle any "big theorems." Instead …

You could keep an eye on the Quaderni del Centro di Studi Grothendieckiani, a biennial refereed journal dedicated to exploring Alexander Grothendieck's mathematical and intellectual legacy. The first …

I will restrict attention to compact level sets $\hat{\Phi}^{-1}(c)$ of $\hat{\Phi}:M\to \mathbb{R}$. Below I sketch a proof of what is essentially a global implicit function theorem in your setting. …

I think you cannot get much more than your condition, at least with respect to the usual structural parameters you mentioned, but it might be studied in another branch of graph theory I am unaware of. …

The equivalence in the last line is not right. To see this, let $S\subseteq V$ be a maximal, linearly independent set, i.e. $$\forall s\in V\setminus S: S\cup \{s\}\;\text{is linearly dependent}.$$ Th …

My comment as an answer: The ODE can be simplified with the substitution $f(t)\mapsto e^{2G(t)}$ and $G'(t)=g(t)$ to read \begin{align} \label{eq:1}\tag{1} y''(t) + g(t) y'(t) + k^2 y(t) = 0 \,. \end{ …

Geometric group theory is a huge subject, and a course that really tried to cover all of it would be too disjointed to be useful. If I were teaching such a course, I would choose a few major theorems …

I know the references for question 4: MR3659421 Schleicher, Dierk; Stoll, Robin Newton's method in practice: Finding all roots of polynomials of degree one million efficiently. Theoret. Comput. Sci. 6 …

Fix $m$, and $a_1,\ldots,a_m$. Let $\mathbf{x}=(x_1,\ldots,x_m)$ and $\mathbf{y}=(y^1_1,\ldots,y^1_{a_1},\ldots,y^m_1,\ldots,y^m_{a_m})$. Consider the set of points $\mathcal{P}\subseteq \mathbb{R}^{m …