297 problems

Katok's conjecture. The collection of ergodic entropies contains

The classical entropy for a quantum-mechanical density matrix on $L^2(\mathbb{R})$ is defined through the associated Husimi function, obtained by evaluating the density matrix on G…

Lieb's generalized Wehrl conjecture. The analogue of Wehrl's conjecture should hold for at least a wide class of Lie groups.

Let $PM(d)$ be the space of primitive majors for degree $d$, and let $\Pi(s_1,\ldots,s_r)$ be a stratum other than $\Pi(d)$ and $\Pi(1,1,\ldots,1)$. Consider the set of primitive m…

Entropy conjecture. The map $\varphi$ minimizes entropy in its homotopy class, and

Sinai's conjecture. For sufficiently large $k$, the standard map $T$ has positive metric entropy with respect to Lebesgue measure.

For a hypersurface $\Sigma$ in Euclidean space $\mathbb{R}^{n+1}$, define its entropy by … Here the supremum is over all $t_0>0$ and $x_0\in\mathbb{R}^{n+1}$. Entropy-minimizing sp…

Let $\Gamma$ be a countable discrete group and let $f\in\mathbf{Z}[\Gamma]$. Let $\boldsymbol{X}_f$ be the algebraic action obtained from the Pontryagin dual of…

Kikuta–Takahashi's conjecture. One has

For a quadratic polynomial, let its core entropy be the entropy of the dynamics on its Hubbard tree, and let the polynomial vary in the space of quadratic polynomials equipped with…

Buzzi's conjecture. The diffeomorphism $f$ has at most countably many ergodic measures of maximal entropy.

Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_i\in[0,1]$, let $\mathbf{p}=(p_1,\ldots,p_n)$, and let $H(\mathbf{p})$ denote the Shannon entropy…

Ecker's monotonicity conjecture. In the case of mean curvature flow for compact embedded hypersurfaces $M_t$ in $\mathbb{R}^{n+1}$, the entropy satisfies

Let $X$ and $Y$ be independent identically distributed log-concave random variables in dimension $1$, and let $h$ denote differential entropy. Ball–Nayar–Tkocz conjecture. The map…

A measure-preserving dynamical system has zero entropy when its Kolmogorov–Sinai entropy is zero. A system is loosely Kronecker when it is loosely Bernoulli; equivalently, in the z…

For a polynomial interval map with real critical points, let an isentropic level set be a set of maps having the same topological entropy. Milnor's monotonicity of entropy conjectu…

Logarithmic convexity conjecture. For every $c\in\mathbb{R}$, $S_{n,c}$ is logarithmically convex; equivalently, $\log S_{n,c}$ is convex on $I_c$.

Let $F_N$ range over probability densities on Kac's sphere, and define the entropy-entropy production ratio by … where $H_N\left(F_N\right)$ is the relative entropy and…

Desvillettes–Villani conjecture. Time oscillations should occur in the evolution of the relative local entropy. This conjecture concerns the long-time behavior of solutions to the…

Entropy flexibility is the property that, for each admissible entropy level, there is an invariant compact set whose restricted system has that entropy. A property is $C^1$-generic…

Off-shell free-energy extremization conjecture. The entropy or central charge of a general class of $d=5,6$ SCFTs compactified on $\mathbb M_4$, with twists parameterized by…

Entropy-volume conjecture. For all $n\geq 3$ and all $\alpha\in(0,1)$,

Luzzi–Marmi conjecture. The entropy function $\alpha\mapsto h(T_\alpha)$ is continuous on the full parameter interval $(0,1)$.

Intrinsic ergodicity conjecture. The answer to the preceding question is positive at least for binary subshifts, and every bounded density shift is intrinsically ergodic.

Let $\mathcal{W}$ be a quantum process with quantum environmental entropy $S_{\operatorname{EC}}$, dispersive entropy $S_{\operatorname{dis}}$, and mixing entropy…