29 problems

Let $\Psi_{2,\alpha}$ denote the one-parameter family of degree-2 solutions of the complex sine-Gordon 2 equation constructed in the paper with the $(1,3)$-configuration. Morse-ind…

Let $P_0$ be the polynomial appearing in the degree-2 vortex solution and let $(u_3,v_3)$ denote the kernel pair at Fourier mode $m=3$. Kernel rationality conjecture. The denominat…

Let $(u,A)^T$ be an entire critical point on $bbmathbb{R}^{n+2}$ satisfying … Write each $x \in \mathbb{R}^{n+2}$ as $x=(y,z) \in \mathbb{R}^2 \times \mathbb{R}^n$, and suppose als…

Let $n$-vortex states $\phi$ be considered for a harmonic potential, with rotation frequency $\Omega\geq 0$ and interaction parameter $a>0$. Instability conjecture. An $n$-vortex w…

Let $\psi:\mathbb{R}^2\to\mathbb{C}$ be a vortex solution of the Ginzburg–Landau equation with degree $n$, and call it radially symmetric when its modulus depends only on the radia…

Consider the two-dimensional Ginzburg–Landau equation and entire solutions $u$ satisfying $|u|\to 1$ at infinity. For each degree $d\in\mathbb{Z}$, let $v_d$ denote a planar vortex…

Let $\mathrm{NLL}(\beta=2n)$ denote the space of soliton states at coupling $\beta=2n$, where the more general formula is obtained from polynomials with distinct roots. Jackiw–Pi d…

Gravitating coupled vortex conjecture. The pair $(\mathbb{P}^1,T)$ admits a solution to the gravitating coupled vortex equations if and only if $T$ is $\sigma$-polystable and the p…

Let $\Sigma$ be a sphere, let $L$ be a line bundle over $\Sigma$, and let $D\in \operatorname{Sym}_n\Sigma$ be a divisor. For small $\varepsilon$, let…

Non-Abelian twisted-curve conjecture. All non-Abelian exotic vortex solutions with $\kappa>0$ on constant-curvature backgrounds arise from this twisted-holomorphic-curve constructi…

Twisted-holomorphic-curve conjecture. All such exotic vortex solutions on constant-curvature backgrounds arise from flat $PSU(N_c+1)$ connections and twisted holomorphic maps into…

Twisted-meromorphic-function conjecture. All of the relevant exotic vortex solutions can be found by this construction from a flat $PSU(2)$ gauge field and a twisted meromorphic fu…

Gauged-holomorphic-map conjecture. All such exotic vortex solutions arise from gauged holomorphic maps into complex space forms of dimension $N$.

Consider the semilocal Abelian case $N_c=1$, $N_f=\lambda$, and more generally take the Chern--Simons level away from $\lambda=N_f$. Anyonic duality conjecture. The vortex theory i…

Consider the semilocal Abelian case $N_c=1$, $N_f=\lambda$, with the vortex theory in the setting described in the source. Fermionic duality conjecture. The vortex theory is dual t…

Let the local case be $N_f=N_c=\lambda$, and let quantum vortices be considered in the corresponding background flux. Fermi-vortex duality conjecture. Quantum vortices are fermions…

Consider the two linked lines obtained as preimages of two distinct regular points on the 2-sphere under the Hopf map applied to a Skyrme field. The underlying field is an original…

Let $X=\mathbb{R}^3\cup\{\infty\}$ and let $U:X\to S^3$ be a map of degree $B$. Interpret $U$ as two vortices in $\psi_1$ and $\psi_2$ of $\boldsymbol{\psi}\in S^3$, with winding n…

The $U(N)$ matrix Chern–Simons model is a gauged quantum-mechanical model describing matrix degrees of freedom for $N$ particles. In non-relativistic Abelian Chern–Simons matter th…

BerRyb13 conjecture. For sufficiently small $\mathbb{\varepsilon}$, the degrees of the hole vortices are the same for $\mathbb{C}$-valued and $\mathbb{S}^1$-valued maps.

Largealpha growth conjecture. There are good reasons to conjecture that

Whittaker-like-state conjecture. The state $|\mathcal{N}_{L,\xi}\rangle$ satisfies the Whittaker-like conditions referred to as. This conjecture proposes that the state created by…

Global sign conjecture. The same sign properties should hold for every $t\in(0,1]$:

Sign conjecture. The radial profiles satisfy

Let $A_+,A_->0$ be fixed and define … For the entire equivariant solution … with $B^2<A_+A_-$, write $f^\infty_\pm(r;B)$ for its component profiles. Monotonicity conjecture. The pr…