54 problems

Let $\mathfrak{g}$ be an infinite simple Lie ring of finite Morley rank, and suppose that its characteristic is sufficiently large. Cherlin–Zilber conjecture for Lie rings. Then…

A definable simple Lie ring of finite Morley rank is a Lie ring $\mathfrak{g}$ definable in a structure of finite Morley rank. Its characteristic is the characteristic associated w…

Let $(G,X)$ be a permutation group of finite Morley rank, with $X$ of rank $r$, and let $F$ be an algebraically closed field. A group action is generically sharply $t$-transitive w…

Let $(G,X)$ be a connected primitive permutation group of finite Morley rank, and write $\operatorname{rk}X=n$. Rank bounds and classification conjecture. … If equality holds, then…

Cherlin-Zilber Algebraicity Conjecture. The group $G$ is isomorphic to a simple algebraic group over an algebraically closed field.

A Moufang set is a permutation structure with associated root groups; a Moufang set is proper when its associated group is not sharply $2$-transitive. For a quadratic Jordan divisi…

Let $n$ be a positive integer, let $H$ be a nonsolvable connected group of finite Morley rank, and suppose that $\operatorname{Alt}(n)$ acts definably and faithfully on $H$ by auto…

Algebraicity Conjecture. Every infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field.

Let $(G,V)$ be a module of finite Morley rank, with $G$ connected and simple modulo its at most finite centre. Suppose that $G$ is transitive on $V\setminus\{0\}$ and generically…

Let $G$ be an infinite simple group of finite Morley rank. Cherlin–Zil'ber algebraicity conjecture. The group $G$ is a linear algebraic group over an algebraically closed field. Th…

Centralizer characterisation conjecture. If $C\cong \operatorname{GL}_2(K)$, then $G\cong \operatorname{PSL}_3(K)$; if $C\cong \operatorname{PSL}_2(K)\times K^*$, then…

Let $G$ be a group of finite Morley rank, let $i\in G$ be an involution, let $C_G(i)$ denote its centralizer, and let $O(C_G(i))$ denote the largest definable connected normal nilp…

Classification conjecture for minimal $p$-algebras. Then

Logarithmic Cherlin–Zilber conjecture. If $p$ is sufficiently large, then $\mathfrak{g}$ is isomorphic to a finite-dimensional Lie algebra over a definable algebraically closed fie…

Let $(G,X)$ be a connected primitive permutation group of finite Morley rank. Generic transitivity conjecture. Almost all such groups have maximum degree of generic transitivity at…

Let $G$ be a connected group of finite Morley rank with finite center $Z(G)$ such that $G/Z(G)$ is simple. Let $G$ act definably and faithfully on an abelian group $V$, transitivel…

Let $(G,X)$ be a connected primitive permutation group of finite Morley rank and regular type. Nonexistence conjecture for primitive groups of regular type. No such group exists. T…

Let $M$ and $S$ be connected groups of finite Morley rank, with $S$ simple. Suppose that $M$ acts faithfully and irreducibly on $S$. Definable-on-simple conjecture. Then $M=S$, and…

For each integer $n\geqslant 1$, let $\tau(n)$ be the maximum degree of generic transitivity of a definably primitive action on a set of Morley rank $n$. Generic transitivity conje…

Strong regular-type nonexistence conjecture. The group $G$ does not have such an action that is generically $2$-transitive.

Regular-type nonexistence conjecture. The group $G$ does not have an action with these properties that is generically $t$-transitive on $X$.

Affine-type classification conjecture. The action is generically sharply $t$-transitive, so in particular $\operatorname{rk}(G)=nt$, and one of the following occurs:

Super tight programme conjecture. The group $G$ is isomorphic to a Chevalley group $X(K)$ over an algebraically closed field $K$ of characteristic different from $2$.

Let $\mathfrak{g}$ be an infinite simple Lie ring of finite Morley rank. Assume that its characteristic is sufficiently large. Large-characteristic conjecture. Then $\mathfrak{g}$…

Simple Lie rings of finite Morley rank are the model-theoretic analogue of the objects classified by the Block–Premet–Strade–Wilson theorem. Classification conjecture. Classifying…