7 problems

Let $U$ be the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over $\mathbb{R}$. The totally nonnegative part of the link…

Let $G/P$ be an arbitrary flag variety, and consider its totally nonnegative part together with the cell decomposition induced by total positivity. Regular CW ball conjecture. The…

Let $Gr(k,n)_{geq 0}$ be the totally nonnegative Grassmannian, stratified by the nonnegative parts of the positroid cells $(\Pi_u^w)_{\geq 0}^\circ$. Postnikov's conjecture. The po…

Let $U$) be the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over $\mathbb{R}$. Let $f_{(i_1,\dots,i_d)}$ be one of the…

A regular CW sphere is a regular CW complex whose underlying space is a sphere, and its $cd$-index is the polynomial $\Phi_P(c,d)$ encoding its flag $f$-vector, with $c$ of degree…

Simple-polytope lower-interval conjecture. For each element $(\sigma,J)\in\mathcal W$, the lower interval $\mathcal W_{\preceq(\sigma,J)}$ is isomorphic to the face lattice of a si…

Lusztig's ball conjecture. The cell decomposition of $(\mathbb{G}/P)_{\geq 0}$ should be a regular CW complex homeomorphic to a ball.