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Not all interesting examples come from algebraic geometry and number theory -- your questions are fairly natural in many other settings. For instance, Conway's game of life qualifies as an answer to y …

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial map …

Start with an edgeless graph on $n$ labeled vertices, and note that the automorphism group is $\Sigma_n$, the symmetric group on $n$ elements. Now imagine that we randomly start throwing in all of the …

Nice question! I ran into a similar problem a few years ago -- even for dimension $10$, the rejection method was annoyingly slow. One of the problems is that such questions straddle at least three hug …

Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the tw …

The question is simple: What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)? I'm not sure what the correct measure on $G_ …

Regarding the question Has this process, or something close to it, been studied before? I was recently made aware of an intriguing approach of Bob Macpherson and his post-doc Ben Schweinhart at …

Okay, so this is nowhere near a complete solution, but this is as far as I got and hopefully someone else sees it from here: It is clear that $P(2,k) = \frac{k-1}{k+1}~$ which is certainly non-decrea …

While the original question regarding permutations is interesting, it is not true that combinatorial Morse functions are hard to construct algorithmically on regular CW complexes. Much work has gone i …