29 problems

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 $v$ be a primal field and let $p$ be the variable obtained from $v$ by a change of variables motivated by a Legendre transform. Consider the resulting dual action and the intro…

Let $U_{\phi}$ be the set appearing in the source, and let $\mu$ be a non-trivial gradient Gibbs measure, meaning that $\mu$-almost surely $Z_{\Lambda}(\phi)<\infty$ for every fini…

Consider perturbed simply attractive potentials, including anisotropic potentials, meaning potentials that are not necessarily isotropic. Let the variational principle be the relat…

Consider a sequence of finite regions converging to a fixed shape, and let $(s,t)$ be the tilt given by the variational principle wherever that tilt is defined. Let $\mu_{s,t}$ be…

Variational principle conjecture. The pressure can be expressed as the supremum of a variational functional depending on these two measures.

Let $\mathcal{A}_s$ be the steady cost, identified in the inviscid setting with the $\mathbb{L}^2$-norm of the convective acceleration. Consider the irrotational inviscid limit of…

Let $\mathcal{A}_s(\boldsymbol{u}):= \mathcal{A}(\boldsymbol{0};\boldsymbol{u})$ denote the steady cost, where $\boldsymbol{u}$ is a flow state and $\boldsymbol{0}$ denotes zero in…

Variational conditional law of large numbers conjecture. For $(x,y)\in\mathbb R_+^2$,

Entropy variational-principle conjecture. Under suitable physical constraints on the refinement parameters, this entropy functional may serve as a variational principle for simulat…

Dong–Qiao's variational-principle conjecture. For every such $(\mathcal X,f)$ and every $r>0$,

Extension conjecture. The Gibbs variational principle remains valid after any of the following minor modifications: replacing spheres by another type of convex body; replacing the…

Uniform bounds conjecture. There exist constants $0<c<C$, depending on all the data, such that $$

Let $(X,d)$ be a compact metric space, let $f:X\to X$ be a homeomorphism, and let $\mathcal{M}(f,X)$ be the invariant Borel probability measures for $f$. For $r>0$, define the $r$-…

Let $(X,d)$ be a compact metric space, let $f:X\to X$ be a continuous self-map, and let $\mathcal{M}(f,X)$ denote the invariant Borel probability measures for $f$. For $r>0$, write…

Direct convergence conjecture. This convergence should also hold without a regularization step, in a direct sense analogous to the doubly periodic height-function convergence theor…

Global minimality conjecture. This minimality extends to all increases in the crack set, not only to regular increments.

Let $P(x,y)$ be the unregularized kernel of the fermionic projector, written as … where $\slashed{p}=\gamma^j p_j$, $\Theta$ is the Heaviside function, and $k(x-y)$ is the Minkowsk…

Let $a_i$ and $F_i$ denote the acceleration and impressed force associated with the $i$-th Cartesian degree of freedom of a mechanical system, with masses $m_i$, and define Gauss's…

Causal triviality conjecture. If the dimension $f$ of the Hilbert space is sufficiently small, then every minimizing measure should be causally trivial. This would reduce the struc…

In the causal action principle, let $\rho$ be a weighted counting measure with $m$ spacetime points on operators of fixed trace, normalized by … Here $n$ is the spin dimension and…

The variational principle for skew Young tableaux. The function $g(t,x)$ maximizes

The paper considers the $1$-finite variational principle ($1$-$\mathrm{FVP}$) and the finite variational principle ($\mathrm{FVP}$) for a space $\mathcal{X}$, formalized over the w…

For $T\geq 1$, let $\mathcal{P}^T$ be the $T$-fold product of the trial distribution space, let $p_T^{(1)}(\underline{\mathsf{x}})$ be the generalized one-letter functional, and le…

Minimizer principle for peakon travelling waves. Peakon travelling waves are the solutions of the minimization problem