5 problems

Affine paving conjecture. There exist affine pavings of ${}^0\operatorname{Hilb}^{(m)}(V)$ and $Z^{(m)}(V)$ by the parts $S_T^{\rm new}$ and $S_{\widehat{T}}$, respectively, such t…

Generalized affine-paving conjecture. If $\mathfrak{o}$ is a total order such that $P^{\mathfrak{o}}(t)=P_V(t)$, then the decomposition of $V$ according to $\mathfrak{o}$ is a gene…

Formal Poincare polynomial conjecture.

Mirkovic–Vilonen cycle and affine-paving conjecture. For every $x\in\mathscr{X}$, there exists $w\in W$ such that $w\cdot\mathscr{X}(\mathrm{Ec}(x))$ is a Mirkovic–Vilonen cycle. M…

Fundamental-domain purity conjecture. The cohomology of $F_{\gamma}$ is pure in the sense of Deligne.