8 problems

Let $\mathcal{C}=(C_j)_{1\le j\le k}$ be a tuple of conjugacy classes of $\operatorname{GL}_n$, defining a star-shaped quiver $Q$, a deformation parameter $\mathbf{q}$, and a dimen…

Let $\kappa=2$, and consider tuples of conjugacy classes for which conditions i) and ii) of the source's genericity theorem hold, but for which the weak Deligne-Simpson problem is…

Questions on negative indices of rigidity. 1) Is the dimension of the variety of $(p+1)$-tuples with non-trivial centralizers always smaller than the expected dimension of the vari…

Nonexistence conjecture. There exist no $(p+1)$-tuples of matrices $A_j$ or $M_j$ satisfying the additive or multiplicative matrix equations, respectively, with trivial centralizer…

Deligne–Simpson conjecture. Conditions i) and ii) from Theorem are necessary and sufficient if $q=1$. For $q>1$, there are cases in which they are sufficient and cases in which the…

Let $\mathbf{S}=(S_i)_{i=1,\ldots,d}$ be a collection of abstract spectral types. A collection is irreducibly realizable if it is the spectral type of an irreducible meromorphic…

Let $G$ be a simple complex group with Lie algebra $\operatorname{\mathfrak{g}}$. Fix a Coxeter $G$-formal type $\mathscr{A}$ of slope $r/h$ with $\gcd(r,h)=1$. A nilpotent orbit t…

Choose a marking of each conjugacy class $\breve{\mathcal{C}}_i\subset \operatorname{GL}(V_i)$ for all $i\in I$. Gluing legs to obtain a supernova graph $\widehat{\Gamma}$ with nod…