97 problems

Consider a Hilbert space ${\mathcal{H}}$ of finite dimension $d=|{\mathcal{H}}|$. Let $\rho$ and $\sigma$ be states on $\mathcal{H}$. Define … The function $f_d(\varepsilon,\nu)$ i…

Let $f$ be a $C^2$-diffeomorphism on a compact Riemannian manifold of dimension greater than one. The map $f$ has the intermediate entropy property when, for every number…

Sinai's conjecture. For every parameter $\lambda\ne 0$, the metric entropy of $f_\lambda$ is positive. This is presented as another famous positive-entropy conjecture; the source d…

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…

Sharp constant conjecture. The sharp value of the universal constant $C_1$ in the Colding–Minicozzi codimension bound is

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

Kikuta–Takahashi's conjecture. One has

Let $Y$ be a smooth projective variety and $f:Y\dashrightarrow Y$ be a rational dominating map. A smooth birational model is a smooth projective variety $\hat Y$ together with a bi…

Completely monotone conjecture. The function $t\longmapsto I(X,t)$ is completely monotonous, that is

Consider the causal model $\mathcal{M}=(\{X,Y\},E,f,X\rightarrow Y,p_{X,E})$, where the discrete random variables $X$ and $Y$ have $n$ states, $E$ is independent of $X$ and has…

The transverse symmetry and absence of correlations reduce the area-density problem to a chiral theory on a lightray, which is mapped by a Cayley transform to a chiral theory on th…

Equipartition duality. Among all such states, the phase-equipartitioned product state, unique when it exists, simultaneously (i) maximizes the entropy, (ii) minimizes every within-…

Flexibility above the Besson–Courtois–Gallot threshold. For every

Entropy sum lower-bound conjecture. The constant $1/8$ can be improved to $1/2$, meaning that

Let $\mathcal N_0^{A_0\to B_0B_1}$ and $\mathcal N_1^{A_1\to B_0B_1}$ be the partially erasing channels defined by fixed states $\sigma_0\forin\fomathcal D(B_0)$ and…

Fix $r>1$, and let $M$ be the underlying compact manifold and $f:M\rightarrow M$ a $\mathcal{C}^r$ local diffeomorphism. Write $\lVert f\rVert_{\mathcal{C}^1}$ for the…

Let $M$ be a closed surface and let $f:M\rightarrow M$ be a $\mathcal{C}^{\infty}$ local diffeomorphism. For an $f$-invariant measure $\mu$, let $\operatorname{HC}(\mu)$ denote the…

Let $Z$ be an $n\times n$ positive semidefinite similarity matrix with $n>0$, symmetric entries in $[0,1]$, and diagonal entries equal to $1$. On the uniform distribution, write LC…

Let $X$ and $Y$ be independent and identically distributed log-concave random vectors taking values in $\mathbb{R}^d$, and let $h$ denote differential entropy. Folklore conjecture.…

Let $\Phi$ be an iterated function system, let $p$ be a probability vector, and let $\mu$ be the corresponding self-similar measure. The entropy rate is … where $X_1,X_2,\ldots$ ar…

Let $P$ be a finite poset, let $h(x)=\mathbb E f(x)$ for a uniformly random linear extension $f$ of $P$, and let $\operatorname{gap}(P)$ be the maximum consecutive gap among the or…

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…

Fix a rational map $R$ over $K$ of degree $d\geq 2$, and for $n\in\mathbb{N}^*$ let $\operatorname{RFix}(R^n)$ be the set of repulsive fixed points of $R^n$. Let $\mu_R$ be a measu…

Let $H_t(\mathcal{G})$ denote the entropy of the random geometric graph distribution at parameter $t$, and let $\bar{p}_{\max}$ be the corresponding edge-probability parameter at w…