17 problems

Let $N>3$, let $\omega\in(0,N)$, and let $\xi\in\Xi_N$. Let $q^{\omega}$ be an action minimizer of $\mathcal{A}_{\omega}$ among all loops in $\Lambda^{D_N}_{\xi}$ satisfying the co…

Let the curves in Figure be energy curves for the problem, with solutions parametrized by their energy and with parameter $\lambda\in\mathbb{R}$. The conjectured joined diagram is…

Let $m$ and $\alpha$ be the parameters defining the weighted Sobolev quotient $\mathcal{S}(m,2,\alpha,\alpha,\infty)$, and suppose that this quotient admits a $0$-th nonsingular mi…

Let $S_{\mu,1,1}$ denote the set of maximizers associated with impulse $\mu$, and let $\mu$ be the impulse parameter. Small-impulse uniqueness conjecture. There exists a universal…

Let $\mathcal{L}$ be the admissible class of Lagrangian objects and let $\mathcal{S}$ be the Solomon functional. A special Lagrangian is a Lagrangian on which the relevant holomorp…

Let $\mathcal{L}$ be the admissible class of Lagrangian currents and let $\mathcal{S}$ be the Solomon functional. Let $\hat{\theta}$ be the phase determined by the relevant Fukaya-…

Let $\mathcal{L}$ be the class of admissible Lagrangian objects and let $\mathcal{S}$ be the Solomon functional. A minimizer is an element $L\in\mathcal{L}$ attaining the infimum o…

Let $\mathcal{L}$ be the class of quantitatively almost calibrated exact unobstructed Lagrangian objects under consideration, and let $L_0\in\mathcal{L}$ be exact, quantitatively a…

Dissipation-minimization conjecture. The parameters $(V,A_s,A_f)$ are chosen in such a way that the dissipation $R$ is minimal. The claim concerns the selection of parameters for a…

Let $d\geqslant2$ and $p>q>1$, and let $I(\lambda)$ be the fixed-mass variational problem … Let $\lambda_c$ denote the critical mass appearing in the variational theory. Uniqueness…

Assume that $\mathcal{K}$ is not central. Let $M einmathbb{N}$ be as in the symmetry condition, and let $(Deltatheta){\min}$ be the minimum total angle compatible with $\mathcal{K}$.…

Absence-of-binding conjecture. For all $2\leq p<3$ and all $N\geq2$, $J(N)$ has no minimiser and

Binding conjecture. For every $N\geq2$ and every $1<p<\min\left(2,1+\frac{2}{d}\right)$, the binding inequalities

Let $s=1$ and consider an obstacle problem of the form of the wave equation with obstacle, with obstacle $g$, and let $u$ be the weak solution obtained through the convex minimizat…

Let $c_{TF}>0$, and for $0<\mu<\frac{15}{64c_{TF}}$ let $Q_\mu$ be the unique positive solution of the associated Euler–Lagrange equation, with mass $M(\mu)$, and let…

Let $(X,\omega)$ be a compact Kähler manifold, let $(\mathcal E^1,d_1)$ be the metric completion of the space of Kähler potentials, and let the extended Mabuchi K-energy be a map…

Let $A$ be a positive definite connection over a compact 4-manifold, and let $S$ be the volume functional on such connections. Positive definite connection deformation conjecture.…