11 problems

Conjectured sharp local tail bound. The same inequality should hold with the power $-1$ in place of $\frac{d}{2}-1$:

Let a Hamming sphere or Hamming ball be a subset of the discrete cube, and let its noise stability be measured at a given noise level among sets of the same cardinality. Hamming-sp…

Let $f$ range over homogeneous polynomials of a fixed degree on the discrete cube, and let $K_s$ denote the Krawchouk polynomial of that degree. Krawchouk extremality conjecture. T…

The sharp stability conjecture. If $\mathcal{F}\subset\mathcal{P}([n])$ and $\mathcal{L}\subset\mathcal{P}([n])$ is as above, then there exists a family…

Let $\{-1,1\}^n$ carry the uniform probability measure, and let the random-walk semigroup on the discrete cube play the role of the Ornstein–Uhlenbeck semigroup. For a non-negative…

Let $n\geq 1$ and $t\geq 0$, let $\mu_t=\frac{1-e^{-t}}{2}\delta_{-1}+\frac{1+e^{-t}}{2}\delta_1$, and let $\mu=\mu_\infty$ be the uniform probability measure on $\{-1,1\}$. Write…

Let $A\subset\{0,1\}^n$ be monotone increasing, with measure $\alpha=|A|/2^n\leq 1/2$. Its edge-boundary is denoted by $\partial A$. Kahn and Kalai's conjecture. For every $L>0$, t…

Boundary excess conjecture. If $A\subset\{0,1\}^n$ has size $k$ and requires at least $l$ additions and at least $l$ deletions to become isomorphic to $C_{n,k}$, then

Talagrand constant-two conjecture. The influences satisfy

Samorodnitsky's conjecture. For every $\delta>0$, there exists $a>0$ such that any $A\subset\{0,1\}^n$ satisfying

Bollobás–Leader–Riordan conjecture. There is a constant $K_N$, depending only on $N$, such that if