10 problems

Let $\Omega_j$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be the…

Strip energy ball comparison conjecture. If

Let $n\geq 2$ and $0<p<n$. Let $K\subset\mathbb{R}^n$ be a compact set with an irreducible group of isometries, let $M$ be a volume-preserving invertible linear map, and write…

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass, and assume that $\partial M$ is empty or minimal. For $V>0$ sufficiently la…

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass. A Bernoulli surface is a surface $S=\partial A$ for a compact set $A$ such…

Let $h$ be a continuous, increasing, concave function defined and positive in some right neighborhood of $0$. For a set $E\subset\mathbb{R}$, define its $h$-volume by … where … Ass…

Let $p,q \in \mathbb{R}_+[x_1,\ldots,x_n]$ be homogeneous real stable polynomials of total degree $d$. For a multi-index $\mu$ with $|\mu|=d$, write $p_\mu$ and $q_\mu$ for the cor…

Consider the three-user semi-symmetric complex Gaussian interference channel … … … where $P_k=P$ for $k=1,2,3$, and write $\phi(g_1)$ and $\phi(g_2)$ for the phases of $g_1$ and…

Let $\Delta$ be the unit disc in the complex plane, let $\mathbb T=\partial\Delta$, and let $E\subseteq\mathbb T$ be closed. For $0<r<1$, write…

Capacity lower-bound conjecture. There exists a constant $C_{a,n}>0$ such that, for every compact non-pluripolar set $E\bigsubset A$,