10 problems

For $k=4$, let $\operatorname{Msp}(n,4)$ denote the maximum spread among all realizable bases for the parameter $n$. Square-root formula conjecture. For every $n\geq6$, … This is a…

Monotonicity conjecture. For each fixed $k$, $\operatorname{Msp}(n,k)$ is a monotonic non-decreasing function of $n$. The conjecture is motivated by numerical tables of maximum spr…

Fix $n$ and $k$, and let $d$ be an exceptional point, meaning a parameter value where several basis elements leave simultaneously during a transition between consecutive realizable…

Symmetry conjecture. For $1\leq i\leq N$,

For fixed $n$ and $k$, an exceptional point is a parameter value at which several basis elements leave simultaneously and the transition to the next realizable basis is exceptional…

Let $I_i$ and $I_{i+1}$ be consecutive realizable bases at a transition point. Simultaneous-change conjecture. If several elements $i_1,i_2,\ldots,i_r$ leave $I_i$ simultaneously,…

Fix $n$ and $k$, and let $I_i$ and $I_{i+1}$ be consecutive realizable bases, with $L(I_i)=[c_i,d_i]$ and $L(I_{i+1})=[c_{i+1},d_{i+1}]$, where $c_{i+1}=d_i$. Minimal-change conjec…

Let $n$ and $k$ be fixed, let $\mathcal{R}=\{I_1,I_2,\ldots,I_N\}$ be the lexicographically ordered set of realizable bases, and let $L:\mathcal{R}\to\mathcal{L}$ map each realizab…

Let $f$ be a Boolean function on $n$ bits, and let $E_{\mathbb{Q}}(f)$ denote its expectation under a distribution $\mathbb{Q}$. For $k\geq 0$, say that most of the Fourier mass of…

Let $K_2$ denote the independence parameter associated with a Boolean function. Consider a Boolean circuit of depth $d$ and size $s$. Linial–Nisan conjecture. Any Boolean circuit o…