7 problems

For $n\in\mathbb{N}$, let $N_n$ be a Poisson random variable with parameter $n$, and let $\mathcal{P}_{N_n,d}^{\bbbeta}$ and $\mathcal{P}_{n,d}^{\bbbeta}$ denote the corresponding…

Let $\bbbeta=(\beta_1,\ldots,\beta_m)$ be the block-parameter vector for the block-beta random polytope $\mathcal{P}_{n,d}^{\bbbeta}$. Negative-parameter extension conjecture. The…

Let $m\in\mathbb{N}$. For $i\in\\{1,\dotsc,m\\}$, let $K_i\subset \mathbb{R}^{d_i}$ be a convex body whose boundary is twice differentiable with positive Gaussian curvature everywh…

Let $A_n$ be a root system with positive roots $\Sigma^+$, Weyl group $W$, and positive Weyl chamber $C^+$. For a Weyl chamber $C$, let $\mathcal{Short}(C)\subset W$ be the set of…

Growth conjecture for primary Carmichael numbers. For sufficiently small $\varepsilon>0$,

Let $f_l$ be the random spherical harmonic and let $V$ be the vector field on $S^2$ considered above. Writing $DF_l$ for the derivative of the map $F_l=(f_l,Vf_l)$ and…

Growth conjecture for $\mathcal{P}_r$. For each $r$, there is a number $\delta_r>0$ such that