10 problems

Uniqueness conjecture for simply connected quadrature domains. Simply connected quadrature domains are uniquely determined by their quadrature function, up to conformal radius.

Let the preceding method produce a quadrature domain $\Om$ with three nodes, and let $T_0>0$ be a time. Consider a Hele–Shaw flow that starts with no liquid at time $T=0$ and has t…

Algebraic Calogero–Moser classification conjecture. The algebraic Calogero–Moser equations exhaust, up to a gauge equivalence, all possible second-order elliptic PDEs whose solutio…

An arclength null quadrature domain is a domain satisfying the arclength null quadrature condition described in the paper. Connectivity conjecture. There exist arclength null quadr…

Quadrature-domain critical-point conjecture. The number of critical points satisfies

Extremal critical-point conjecture. For every $n\geq 2$, and for each $k=1,2,\ldots,n+1$, such a domain exists. The bound is attained for $k=n+1$ by the extremal construction descr…

A 1-point quadrature domain is a domain whose Bergman span contains all holomorphic polynomials. If $f:D_1\to D_2$ is a polynomial biholomorphism with polynomial inverse and consta…

Let $n>2$. A quadrature domain in $\mathbb{R}^n$ is a domain satisfying a quadrature identity for the relevant class of functions, and an algebraic domain is one whose boundary is…

Let $\Omega$ be a subharmonic quadrature domain as above, with its associated function $u$ and boundary decomposition as in the preceding setup. Subharmonic quadrature-domain conje…

Three-dimensional Neumann-oval conjecture. $W$ can be analytically continued throughout $\Omega\setminus B$. This is a proposed higher-dimensional analogue of the Schwarz potential…