15 problems

Let $X$ and $Y$ be smooth varieties, let $\pi:X\longrightarrow Y$ be a morphism, and fix a dimension $k$. Consider the $\mathbb{R}$-vector space generated by pseudo-effective $k$-d…

Let $Sigma={a,b}$ be the binary alphabet. A morphism $h$ on $Sigma$ is abelian 5-power free if its iterates avoid abelian fifth powers, and its fixed point is the infinite word gen…

Let $Sigma$ be a finite alphabet with fewer than six letters, and let a morphism on $Sigma$ be abelian power-free if it avoids abelian powers. Carpi's set of abelian power-free mor…

Let $Sigma$ be a finite alphabet, and let a morphism on $Sigma$ be called abelian power-free when its images avoid abelian powers of the relevant exponent. Carpi's set of condition…

A rational homogeneous space is a homogeneous projective variety associated with a semisimple algebraic group, and its semisimple rank is the rank of that group. Kumar's conjecture…

The explicit limit conjecture for $L(2)$.

Let $L(w)$ denote the lexicographically least infinite word on $\mathbb{N}$ beginning with $w$ whose only square factors are contained in the prefix $w$. Let $L(\varepsilon)$ be th…

Pseudopalindromic-closure morphism conjecture. Only morphisms of the form

Antipalindromic fixed-point classification conjecture. The morphism $\varphi$ or its square $\varphi^2$ is conjugated to a morphism in

Antipalindromic morphism classification conjecture. The two classes of morphisms introduced in the paper contain, up to conjugation, all morphisms that generate infinite words with…

Let $T$ be the alphabet, and let an $n$-uniform squarefree morphism be a morphism $h:T^*\to T^*$ for which every image $h(a)$ has length $n$ and which maps squarefree words to squa…

Reversal-symmetry conjecture. The language $\mathcal L({\mathbf u})$ is closed under reversal if and only if ${\mathbf u}$ is palindromic.

Class $P'$ conjecture. There exists a morphism of class $P'$ such that its fixed point has the same language as ${\mathbf u}$.

Zero defect conjecture.

Let $X$ be a projective variety of dimension $n$, let $\pi:X\to Z$ be a surjective morphism of projective varieties of relative dimension $e$, and let…