10 problems

Let $\mathcal Q$ be the space of paths used to parameterize overlaps, let $q=(q_1,q_2)\in\mathcal Q^2$, and let $\psi_2:\mathcal Q^2\to\mathbb R$ be the initial free-energy functio…

Let … where the $U_j(x)$ are fixed non-random potentials of class $C^1$ satisfying condition, and $(a_j^\omega(t),1\leq j\leq N)$ is a realization of a stationary vector-valued ran…

Generic finite-graph conjecture. For a generic such function $A(x)$, the closure of the singular set of $u$ is a finite graph.

Consider the setting of Theorems --, with the convergence rates established there. Optimality conjecture. The convergence rates obtained therein are optimal up to slowly varying fa…

Parisi PDE conjecture. The enriched free energy $F_N$ should converge to a function $f:\mathbb{R}_+\times\mathcal Q^2\to\mathbb{R}$ solving

Consider the viscous Hamilton-Jacobi equation … where $\Omega$ is a domain, $\gamma>1$, and $f\in L^q(\Omega)$ with $q>1$. Write $\gamma'=\frac{\gamma}{\gamma-1}$ for the conjugate…

Stochastic homogenization conjecture. The Hamiltonian $H_{\ell}(p,x,\omega)$ is stochastically homogenizable.

Equal minimal subgradient norm conjecture. If

Let $H(x,p,t)$ be a Hamiltonian satisfying $H(x,p,t)\in C^2$ and convexity with respect to $p$, and let (A5) denote the assumption referenced in the source. Let $\varphi$ be the vi…

Constrained-problem construction conjecture. First solve the unconstrained problem using the formula above. Then, whenever $\min \tilde u\leq 0$, replace $\tilde u$ by $v^2/2$ on t…