8 problems

Derivative probability-ratio conjecture. There exists a constant $\theta\in(0,d/2)$ such that

Positive-drift conjecture. There exists $\theta_k\in(0,1)$ such that, for most reasonable $k$-edge sets $S$ and every positive $m$,

Nonnegative-expectation conjecture. For most sets $S$, the expected value of the higher-order derivative is non-negative:

Derivative-expectation equivalence conjecture. There exists a constant $C>1$ and a family $\mathcal F$ containing $S$, with $|\mathcal F|\geq Cn^{dk}$, such that for every…

Second-order mass conjecture. There exists a constant $C$ independent of $N$ such that

Comparability conjecture. There exists a constant $C$ independent of $n$ such that

Sharp derivative-bound conjecture. The binomial coefficients $\binom{k-2}{\lceil (k-2)/2\rceil}$ are the actual bounds for all $k$; that is, the examples described above attain the…

Small-interaction conjecture. We conjecture that $\mathbb E[N_2]$ and the $L^2$ norms $\|\partial_M f\|_2$ are small, especially in dimensions $d\geq 3$, where the decay could pote…