Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$ Let us say that two copies of $H$ meet nicely if they intersect in exactly 6 points e.g. as the tw...

Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$ Let us say that two copies of $H$ meet nicely if they intersect in exactly 6 points e.g. as the tw...

I asked this question at math.SE a couple of months ago and only got a partial answer, so I thought I would try here. It is known that, for $n \geq 5$, it is possible to partition the integers $\{1, 2, \ldots, n\}$ into two disjoint subsets such that the product of the elements in one set equa...

We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$. I think we can canonically linearize the action of $T$ on $-K_X$. Is anything known about the GIT quotient for this action? Is the semi-sta...

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following properties: i) for any topological space $S\in\operatorname{Top}$, there exists another topological spa...

In the paper Anti-de Sitter space, squashed and stretched Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up Berger spheres, it seems what is meant with "squashing" in the case of $S^3$ is the following: Take the Lie algebra $L$ spanned by $z_1$,$z...

I am trying to parametrize the collection of $d\times m$ real matrices quotient $d\times d$ orthogonal matrices. Formally, define $\sim$ on $\mathbb{R}^{d\times m}$ by $X\sim Y$ if there exists an orthogonal $d\times d$ matrix $Q$ such that $QX = Y$. Then $\sim$ is an equivalence relation on $\ma...

I asked this on the math.stackexchange forum about a week ago but did not get any answers so I figured I might try it here as well. This is a straight up copy paste from my question here. I am interested in the behavior of the quotient semi-metric on geodesic spaces, i.e. length spaces where the...

Suppose that $G$ is a reductive algebraic group acting on a smooth variety $X$, and that the action has finite stabilizers. When is the action of $G$ on $X$ proper? What is an example where the action is not proper? I am aware of a similar statement which is Proposition 0.8 in Mumford's GIT, wh...

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. For $1<p<\infty$ consider the $p$-th power Bochner Integrable functions which takes values in $X$ and defined on the unit interval $[0,1]$, denoted by $L_p(I,X)$. It is clear that $L_p(I,Y)$ is a closed subspace of $L_p(I,X)$. It is w...

Is it possible to take a data set of points in R^3 (specifically, points in the xy-plane) and convert them to a surface of a sphere? Is there any mathematical formula for this? After the suggestion of @ManfredWeis, I looked for stereographic projections, and here is a Cartesian grid on the plane...