8 problems

Let $(\overline{X},\bar{g})$ be a smooth compact Riemannian manifold with boundary, and let $\Theta_n(\overline{X},[\bar{g}])$ denote the supremum of the isoperimetric ratio over s…

Consider the non-cutoff Boltzmann collision operator … where $B(\lvert v-v_*\rvert,\sigma)=\Phi(\lvert v-v_*\rvert)b(\cos\theta)$,…

Let $n$ be the boundary dimension, let $\gamma\in(0,\frac{1}{2}]$ with $n>2\gamma$, and let $(\overline{X},\bar{g})$ be a smooth compactification that is not conformally diffeomorp…

Spectral sign-transform inequality conjecture. For $s\in(1,\frac32)$, the analogous inequalities should hold for spectral quadratic forms. The corresponding inequality has been pro…

Let $\theta$ denote the exponent regulating the reservoir strength $\kappa N^{-\theta}$, and let $\rho_t^\theta$ denote the macroscopic density profile. The profile is governed by…

Let $s>0$ be nonintegral, let $u\in H^s(\mathbb{R}^n)$ change sign, and assume that $u^+$ and $u^-$ belong to $H^s(\mathbb{R}^n)$. Write $\lfloor s\rfloor$ for the integer part of…

Fractional Lieb-Thirring coupling conjecture. The optimal constant also satisfies

Let $S^{2n+1}$ be the CR sphere, let $Q=2(n+1)$, and let $\mathcal P_\gamma$ denote the CR fractional power of the sub-Laplacian on the sphere. For $\gamma$ in the range under cons…