14 problems

Entropy power conjecture. The entropy power should satisfy

Gaussian optimality conjecture. This quantity is exactly minimized, among probability measures with the fixed variance, by Gaussian measures.

Gaussian completely monotone conjecture. The entropy along the heat flow should satisfy

Let $\beta>1/2$, let $w$ lie in the unit disk, and let $\widetilde T_t$ be the modified transport map describing the conjectured motion of an individual zero after it reaches the r…

Modified transport-map conjecture. In the hydrodynamic limit, the maps $\widetilde T_t$ describe the motion of individual zeros. The proposition preceding the conjecture rigorously…

Let $T_t:\mathbb{C}\to\mathbb{C}$ be the deterministic transport maps associated with the hydrodynamic limit, let $t_{\mathrm{sing}}$ be the first singular time, and let $z_j(t)$ d…

Let $z_j(t)$ denote a zero of a heat-evolved polynomial at time $t$, let $w$ be a location of a zero at time $0$, and let $T_t:\mathbb{C}\to\mathbb{C}$ be deterministic transport m…

Let $\nu$ be a rotationally invariant probability measure on $\mathbb{C}\backslash\{0\}$ satisfying $\int_{0}^{\varepsilon} \nu(\{|z| < r\})r^{-1}\,\operatorname{d}r<\infty$ for so…

Let $t\to\mathcal E_t$ denote the elliptic laws, where … and let $P_n$ be the corresponding characteristic polynomials. Hall–Ho heat-flow conjecture. The transition from the circul…

Hall–Ho straight-line conjecture. The zeros tend to evolve along straight lines according to

Real-time global existence conjecture. If $\tau$ is real, then $z^{a}(\tau)$ is well defined as a real analytic function of $\tau$ for every $\tau\in(-1,1)$.

A half-harmonic map is a map $u:\mathbb{R}\to\mathbb{S}$ solving the half-harmonic map equation, and its heat flow is the corresponding half-harmonic map heat evolution. Sire's con…

$T^{-1}$ convergence-rate conjecture. Under the hypothesis of this section, the convergence rate of the entropic interpolation to the heat flow is $T^{-1}$, in the sense of the dis…

Heat-flow coincidence conjecture. The heat flow semigroup $(T_t)$ defined by the paper's formula for $F:=P_{(D)}$ coincides with the heat flow constructed by Sturm.