@BalarkaSen @none @MikeMiller there are no nontrivial finite-dimensional linear representations of $\mathrm{Homeo}^+(S^1)$ with coefficients in any field. Proof: $\mathrm{Homeo}^+(S^1)$ is simple (cf. https://math.unice.fr/~emiliton/cerclesursurfaces.pdf), thus any nontrivial representation is faithful. But $\mathrm{Homeo}^+(S^1)$ contains as a subgroup the Thompson group $F$ (see https://en.wikipedia.org/wiki/Thompson_groups) which violates the Tits alternative (https://en.wikipedia.org/wiki/Tits_alternative), so $F$ can't be embedded into a general linear group
I think the content is a "fracture lemma" which says given an open cover, you may write every homeomorphism as a product of terms which are supported in each chart in the open cover
@Eran Easier. Try $A_4$, similar idea. (Do you know that $A_4$ is the group of symmetries of a regular tetrahedron? That helps see the elements and subgroups a bit.)
@Eran: Sure, your example is fine. But my suggestion doesn't need you to appeal to any big theorems. You can see bare-hands that there's no subgroup of order $6$ in $A_4$.
Here is a sketch of a start of a definition of category.
It uses definitions found in set so we import that. In that file you'll find $\text{cat} \ \textbf{Set}$ defined and all the basic set operations.
Anytime you compile a diagram and a Node is titled "Definition:..." then that triggers an ...
polytope problem of the day: I start with a convex hull in R^n and intersect with an affine subspace. the result will itself be a convex hull. My task: figure out how to relate (computationally) the vertices of the new convex hull to those of the old
Completely unrelated but I answered a set theory question on MSE yesterday and the same question is in this week problem set (which was published today)
There is a paper, I think it is called Recognizing hyperbolic 3-manifolds or something like that, and towards the beginning it has a section on hyperbolic spaces and like 25 or so lemmas listed, then they prove the 25 lemmas. Working through them as exercises should give you a pretty good intuition for many aspects of hyperbolic spaces @AlessandroCodenotti
simpler question which I'm realizing is all I actually want: Suppose I've got the n-simplex in R^n. Then any point in the interior of the hull can be written uniquely as a convex combination of the n vertices.
However, if I add and extend the hull to contain this point, then there will be no unique convex combination
I'm trying to figure out how to make that idea more precise: given a point in a convex hull, characterize the convex combinations of the vertices which generate this point
@AlessandroCodenotti The On the fundamental groups of one-dimensional spaces has a cool example of two contractible spaces whose wedge (glue at a point) is not contractible. (I feel like I have mentioned this to you before...)
Yah, it isn't the type of answer that I would think would get so many votes (also the question got more votes than I would expect). Although maybe a lot more people are interested in combinatorics/graph theory than I expected and just every question is like that...
@TedShifrin @BalarkaSen The statements regarding conjugations are as follows.
1. Any orientation-preserving homeomorphism of $S^1$ (call this a circle map) with irrational rotation number $\rho$ is semi-conjugated to a rotation of angle $\rho$.
1. Let $f$ be a $C^2$ circle map with an irrational rotation number $\rho$, then $f$ is conjugated to the rotation of angle $\rho$.
2. There exists a $C^1$ circle map which is not conjugate to any rotation (Denjoy counterexample).
@MatheinBoulomenos thank you for your proof regarding the nonexistence of representations. It's too high powered for me for sure, but I'll keep it in mind for the future.
Here is a sketch of a start of a definition of category.
It uses definitions found in set so we import that. In that file you'll find $\text{cat} \ \textbf{Set}$ defined and all the basic set operations.
Anytime you compile a diagram and a Node is titled "Definition:..." then that triggers an ...
