97 problems

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 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…

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…

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…

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…

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…

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…

Let $\mathcal H$ be the Hénon family $F_{a,b}(x,y)=(x^2+a-by,x)$ with $a,b\in\mathbb R$, and let $\partial_{\operatorname{chaos}}$ denote the boundary in $\mathcal H$ of the set of…

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 $r$-…

For $p>0$, let ${\mathbb{S}}^{n-1}_p=\{x\in\mathbb{R}^n:\|x\|_p=1\}$, where $\|x\|_p=(\sum_{i=1}^n|x_i|^p)^{1/p}$. Let $\mathbf{P}$ be the cone measure on ${\mathbb{S}}^{n-1}_p$, a…

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…

Let $A\subseteq\mathbb{N}$, let $1_A$ be its indicator function, let $\AE^*(A)$ denote the dual entropy of $A$, and let $\mu$ be the Möbius function. Zero dual-entropy Möbius disjo…