3 problems

Let $x\in \mathcal{S}^{2}$ and, for $t\in \mathbb{R}$, let $\mathcal{L}_{l;x}^{t}$ denote the random length of the unique component of $f_l^{-1}(t)$ that contains $x$ inside, or…

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…

Let $f_{\ell}$ be the random spherical harmonic on $\mathbb{S}^{2}$, let $\mathcal{N}_{\ell}^{c}(I)$ denote the number of its critical points whose values lie in an interval $I$, l…