11 problems

Let $K$ be a field of characteristic $0$, let $n>2$, and let $M_n(K)$ be the algebra of $n\times n$ matrices. Write $s_{2n}$ for the standard polynomial of degree $2n$. Degree-…

Regev's conjecture. For every PI-algebra $A$, there exist $g\in \frac{1}{2}{\mathbb Z}$, $p\in {\mathbb N}$, and $\lambda\in {\mathbb R}_{>0}$ such that

Let $A$ be a finitely generated PI-algebra over the field of rational numbers, and let $J$ be its Jacobson radical. Representability conjecture. If … then $A$ is representable. Thi…

Let $A$ be a Jordan PI-algebra, and let its codimension growth be measured by the sequence of ordinary codimensions. The scale 2 referred to in the paper is the scale for codimensi…

Let $L$ be a finitely generated Lie PI-algebra. Write $\gamma_L(n)$ for its growth function, and let $\ln^{(q)}n$ denote the $q$-fold iterated natural logarithm. Intermediate-growt…

Let $A$ be an algebra, and let the complexity of $A$ be its PI-degree, denoted by $\operatorname{PIdeg}(A)$. Consider all words in the generators of $A$ having length not exceeding…

Let $F$ be an algebraically closed field of characteristic zero. Let $A$ be a finite-dimensional basic algebra with Wedderburn–Malcev decomposition … where $J(A)$ is the Jacobson r…

Volichenko's conjecture. Every PI-algebra over a field of positive characteristic satisfies the symmetric identity $s_d=0$ for some $d$.

Let $W$ be an algebra over an algebraically closed field $F$ of characteristic zero satisfying a polynomial identity of degree $n$. Suppose that $W$ is nondegenerately graded by a…

Let $K$ be the ground field, let $E$ denote the Grassmann algebra, and let $R$ be a PI-algebra with … where each $R_i$ is one of $K$, $E$, $M_2(K)$, or $E\otimes_K E$, and at least…

Let $K$ be the underlying field, let $M_k(K)$ denote the algebra of $k\times k$ matrices over $K$, and let $c(M_k(K),t)$ be its codimension series. For $k=2$, this series is algebr…