9 problems

Let $n\geq m$, let $\mathcal{L}$ be a $2mn$-dimensional lattice in $M_n(\mathbb{C})$, and suppose that … Here $\Delta_m(\mathcal{L})$ denotes the infimum of the products of the sma…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of finite GK dimension. Let $Q(A)$ be its quotient division algebra, and let $D$ be a division s…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension $d$. A chain … of division subalgebras has stratiform length $m$ when each $D_i$…

Self-similarity conjecture. Assume that $n\geq 2$ and $p\geq d(n^2-1)$. Then no open subgroup of $SL_1^{ne}(D)$ is self-similar.

Let ${\mathcal D}$ be an index $n$ ${\mathbb Q}$-central division algebra, let $\Lambda\subset {\mathcal D}$ be an order, and let $\psi_{reg}$ denote its regular representation in…

Let $A$ be a finite-dimensional division algebra over $\mathbb{Q}$, and let $\Gamma_1,\Gamma_2$ be finitely generated semigroups of the multiplicative group $A^\times$. Fix…

Let $d$ be a positive integer, let $F$ be an infinite field of characteristic $0$ containing a primitive $d$th root of unity, and let … be a division algebra, where each factor is…

Let $(X,(U_x)_{x\in X})$ be a proper Moufang set, meaning that its little projective group is not sharply $2$-transitive, and suppose that each root group $U_x$ is abelian. A quadr…

Let $D$ be a central division $F$-algebra, and let $M_{m,D}$ denote its upper motive associated with the generalized Severi–Brauer variety $X(p^m,D)$. Let $K/F$ be a field extensio…