6 problems

Projective fine-moduli-space conjecture. The only fine moduli spaces that are projective are the moduli varieties of stable sheaves, when $r$, $c_1$, and $\chi$ are pairwise relati…

Let $d$ be a positive integer and let $\hat\Omega_d^{\mathbb P^2}$ be the shifted Poincaré polynomial associated with the moduli space of degree-$d$ one-dimensional sheaves on…

Leading Betti-number expansion conjecture. One has

Fix coprime integers $r,\chi$ with $r\geq 3$. Let $M_{r,\chi}$ be the moduli space of one-dimensional stable sheaves $F$ on $\mathbb P^2$ with $[\operatorname{supp}(F)]=rH$ and…

Let $\Lambda_k(\mathbb{RP}^2)$ denote the supremum of the normalized $k$-th Laplace eigenvalue over metrics on the real projective plane. A sequence of metrics is said to degenerat…

Takahashi's conjecture. For all $d\geq 1$,