MathOverflow

This is a very basic question on sheaf theory.
Let $f\colon X\to Y$ be a continuous map of topological spaces. Let $\mathcal{O}_Y$ be sheaf of commutative unital rings on $Y$. Let $F$ be sheaf of $f^{-1}(\mathcal{O}_Y)$-modules. Is it true that the derived functors $R^if_*(F)$ in the category of...

Let $\mu_0$ be a log-concave distribution on $\mathbb R^d$ and let $f:\mathbb R^d \to \mathbb R$ be $C^2$. Let $x_0$ be sampled uniformly at random from a log-concave distribution $\mu_0$, meaning that $\mu_0$ has density of the form $p_0(x) = e^{-U...

Consider the following question:
"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometricallyimmersed in three dimensional Euclidean space$(\mathbb{R}^3, g_{flat})$?"
I believe the answer to this question is no. Can someone give me a referencefor this theorem (in particular I want to look at the details of theproof and understand why this is not possible)....

Despite Timothy Gowers's relief that this recent AI solution of the Erdos Unit Distance Problem only involves a counterexample and not a proof in the positive (see the companion document in the link above), it seems appropriate for the community to weigh...