14
votes

### Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space

Probably, you can find a discussion of this in Thurston's notes on hyperbolic 3-manifolds, or maybe some of the expositions by his students.
However, what you are asking for is actually pretty simple: ...

12
votes

Accepted

### On the origin of a fundamental theorem of additive number theory

I shared the link to this thread with several colleagues, inviting them to contribute to the discussion. Notably, I received a response from Melvyn Nathanson himself, most of which is reproduced below ...

8
votes

Accepted

### Decimal expansion definition of real numbers, constructively

We have a well defined map from decimal expansions to Cauchy real numbers, so by taking the image factorisation of this map, we can always quotient out the set of decimal expansions to get a subobject ...

8
votes

### Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space

This holds more generally in higher dimension too, and I needed similar references about this basic fact of the so-called AdS/CFT correspondence: global conformal maps of the sphere $S^n$ are in one-...

8
votes

### Clausen's modified Hodge Conjecture

In June 2024, Dustin gave a four-hour lecture series in Copenhagen, primarily aimed at explaining and motivating this modified Hodge conjecture. This was part of a "Master class" on ...

7
votes

### Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space

MR0725161 Ahlfors, Lars V. MÃ¶bius transformations in several dimensions, Minneapolis, MN, 1981.

7
votes

Accepted

### Reference for homotopy groups of filtered homotopy colimits

Here's one way to get this out of the literature:
By [Lurie, Higher topos theory, Prop. 5.3.3.3], for filtered $I$ we have that the colimit functor $\mathrm{Fun}(I,\mathcal{S})\to \mathcal{S}$ ...

6
votes

Accepted

### Number of distinct higher dimensional integer partitions

For distinct part plane partitions (called strict plane partitions $\mathcal{SP}$ in the references below), MacMahon's formula
$$ \sum_{\pi \in \mathcal{P}} q^{|\pi|} = \prod_{n \ge 1} \left( \frac{1}{...

6
votes

### About $CW(512,16^2)$

The posed question is very close to questions on the existence of circulant Hadamard matrices and the nonexistence techniques of that theory should suffice.
Suppose $W$ is a circulant weighing matrix $...

4
votes

### Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$

I think that statement is false.
Let $R=\mathbb{Z}[\varepsilon]/\varepsilon^2$ and let $G_1=\mathbb{Z}/2\times 1$ and $G_2=1\times \mathbb{Z}/2$ and $G=\mathbb{Z}/2\times\mathbb{Z}/2$. Let each $G_i$ ...

4
votes

Accepted

### What conditions on the rate matrix $Q$ ensure unique convergence in continuous-time Markov chains?

For continuous time Markov Chains on finite state spaces, see Chapter 20 of "Markov Chains and Mixing Times, second edition", for instance in Theorems 20.1 and 20.3.
A short summary is that ...

3
votes

### Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space

Another reference which takes a very explicit point of view (i.e. entirely matrix calculations and explicit circle inversions) is Beardon's The Geometry of Discrete Groups, which deduces the ...

2
votes

### Calderon-Zygmund/$L^p$ estimates for the linear heat equation

This is not a complete answer.
I've been searching for this kind of reference few months ago (see this post of mine).
Here is what I have collected.
In these notes you have a multiplier approach (...

1
vote

### Orthogonal projection onto cones in inner product spaces

For a given Hermitian $A$, the matrix $A^+$ is the positive semi-definite matrix $X$ that minimizes the Frobenius norm $\| X-A\|_{\rm F}$, as well as the spectral norm $\| X-A\|_2$, see section 8.1.1, ...

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