34 problems

Let $\mathbb D$ be the unit disk, let $b>0$ be the magnetic field strength, and let $\lambda_1(b)$ denote the lowest eigenvalue of the magnetic Neumann Laplacian on $\mathbb D$. Le…

Thermodynamic critical-temperature conjecture. There is a critical temperature $T_c(\lambda,\Omega)>0$ such that, for temperatures $T\geq T_c(\lambda,\Omega)$, the normal-state spi…

Let $b_2$ denote the lower boundary of the surface-superconductivity regime, let $b_3$ denote its upper boundary for smooth domains, and let $\Theta_0$ be the relevant spectral con…

Let $h_a$ denote the bifurcation field for asymmetric solutions and $h_s$ the bifurcation field for symmetric solutions, and consider parameters $(\kappa,d)$ for which asymmetric s…

Aftalion's conjecture. Asymmetric solutions can have lower energy than the symmetric solutions.

Marcus's conjecture. A minimizer of $\mathcal G$ is probably a symmetric solution satisfying (1.2)–(1.6). The source leaves open whether asymmetric solutions may also exist as mini…

Let $\kappa$ be the parameter of the type-I superconductivity model. Define the one-dimensional transition energy by … Let … and let $\sigma_\kappa$ denote the corresponding two-di…

Global monotonicity conjecture. The map

Critical-temperature upper-bound conjecture. There is a critical temperature $T_c(\lambda,\Omega)>0$ such that, for all $\Omega>0$ and $\lambda>0$,

Graph conjecture. All $\tilde{\mathscr{C}}_c^{(N)}$ are graphs over the $\lambda$ axis for general $N\in\mathbb{N}$.

Let $H({\mathbf S}|{\mathbf N})$ be the condensation-energy functional for an admissible spin chain ${\mathbf S}$, where ${\mathbf N}$ denotes the normal-state spin chain, and let…

The model describes dressed bound fermion pairs with total quasi-momentum $k\in\mathbb{T}^{2}$, arising as eigenstates associated with the eigenvalue $\mathrm{E}(\mathrm{U},k)$ and…

Consider the momentum distribution of the ground state of the interacting Fermi gas, and let the Kohn–Luttinger effect denote the superconducting instability expected to smooth the…

The experiments concern the angle dependence of the Josephson coupling between twisted $d$-wave superconductors: earlier measurements found no expected angle dependence, whereas an…

Quarter-space asymptotics conjecture. In dimension $d=3$, the three-dimensional analogues of $L_1$ and $L_2$ are of equal order and converge to finite numbers as $lambda\to0$.

Let $\rho\in[0,1]$, and consider the $U(1)\times\mathbb{Z}_2$ Hamiltonian … where $\rho_\pm=1\pm\rho$, $T_c$ is the superconducting transition temperature, and $T_\text{TRSB}$ is t…

Consider a class of Hamiltonians with $U(1)\times \mathbb{Z}_2$ symmetry, where $T_c$ denotes the superconducting transition temperature and $T_\text{TRSB}$ the time-reversal symme…

The Hubbard model is an extended lattice model in one, two, or three dimensions used as a paradigm for strongly correlated systems; its two-dimensional version is the 2D Hubbard mo…

First Borel-plane singularity conjecture. The first singularity in the Borel plane of the coupling constant is given by $A$.

For $0<\alpha\leq\pi$, let … and let $\operatorname{P}_{1,U_\alpha}=-(\nabla-i\mathbf{A}_0)^2$ be the magnetic Neumann Laplacian on $U_\alpha$. Write…

Let $C^{j\to}_{1,1}(\xi)$ be the scale-$j$ contribution to the two-point function, with $\xi=(\tau,\boldsymbol{x})$, and let $\mu$ be the parameter appearing in the scale bounds. T…

Let $k\in\mathbb{Z}$ denote a Fourier mode in the angular variable, and let the infrared limit mean the limit toward low momentum or energy. The $k=0$ mode is called the $s$-wave.…

One-dimensional reduction conjecture. Under these assumptions,

Let $T_c(b)$ be the largest temperature below which the normal solution is energetically unstable under general $al$-perturbations, and let $T_c'(b)$ be the smallest temperature ab…

Boundary-layer current-conversion conjecture. The normal current $-\sigma\nabla\phi$ turns into the superconducting current $\rho_{out}^2\nabla\chi_{out}$ within a thin one-dimensi…