7 problems

Shapiro's conjecture. $(A,\sigma)$ admits a decomposition as a tensor product of $r$ quaternion algebras with involution in which each quaternion algebra is split.

Let $(R,*)$ be an involutive $A$-algebra, where $A$ is a commutative $\mathbb Z[\tfrac{1}{2}]$-algebra. A weak symplectic determinant law consists of a pair $(D,P)$, where…

Let $\alpha\geq 1$ be a real number. An algebra with involution $A_\alpha$ has a $*$-PI-exponent, when the limit exists, defined by … Realization conjecture for $*$-PI-exponents. F…

Let $A$ be a finite-dimensional algebra with involution $*\colon A\to A$. Its $*$-codimensions are denoted by $c_n^*(A)$, and when the limit exists its $*$-PI-exponent is … Existen…

Pfister involution conjecture.

Let $(A,\sigma)$ be an algebra with involution, and let $J(\varphi_\sigma)$ denote the $J$-invariant of its adjoint quadratic form over the function field of the Severi–Brauer vari…

Procesi–Schacher conjecture. Every totally $s$-positive element of $A$ is a sum of hermitian squares. Equivalently, the trace of a hermitian square is always a sum of hermitian squ…