10 problems

Nash's conjecture. The number of irreducible components of $X_{\infty}^{\operatorname{Sing}(X)}$ is at most the number of essential irreducible components of the exceptional locus…

Kronheimer–Ricci-flat coincidence conjecture. The two metrics are different on $Y^\Gamma_{[3]}$, but they coincide on the exceptional divisor $\mathcal{ED}$.

P=W for resolution. The lift satisfies, for every $k$,

Nonexistence conjecture. There do not exist canonical small resolutions for symmetric rank loci.

Let $X$ be a compact Riemann surface, let $G$ be a complex reductive algebraic group, and choose resolutions of singularities … Let…

Resolution conjecture. Every generalized $n$-manifold for $n\geq 3$ is resolvable.

Let $\mathfrak{g}_{\mathbb{R}}$ be a real form with maximal compact subgroup $K_\theta$, let $\mathfrak{a}$ be a Cartan subspace, and let $W(\mathfrak{a})$ be its Weyl group. Real…

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\operatorname{Hilb}(\mathfrak{g})$ be its generalized Hilbert scheme with Chow morphism $ch$ to $\mathfrak{h}^2/W$. Chow-re…

Sharpness conjecture. For any wild quiver $Q$, there exist dimension vectors $\gamma$ and $\alpha$ such that

Let $(\mathrm{G},V)$ be a prehomogeneous vector space, let $Z\subset V$ be the closure of an orbit of $\mathrm{G}$, and write $Z_{\mathrm{sing}}$ for its singular locus. For an int…