14 problems

Let $m,p>1$, let $\operatorname{Grass}(p,m+p)$ be the Grassmannian, and let $\alpha^1,\ldots,\alpha^n$ be Schubert data. A real upper-triangular matrix with diagonal entries $1$ is…

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $(w_1,\dotsc,w_m)$ be a Schubert problem for $\mathbb{F}\ell(\alpha;n)$. For each permutation $w_i$, let…

Let $\gamma$ be the rational normal curve, and for a set $S_i=\{s_{i1},\dotsc,s_{in}\}$ of $n$ distinct points of $\mathbb{R}\mathbb{P}^1$, let $F_\bullet(S_i)$ be the flag whose…

Let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$, let $\delta(w_i)$ be the unique descent of $w_i$, and let $\Delta_w(t_1,\dotsc,t_m)$ be it…

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$. Let…

Let $a=(a_1<\dotsb<a_k)$ and $n$ be positive integers with $a_k<n$. Let $W^a$ index Schubert varieties on $\mathbb F\ell(a;n)$, and let $w_1,\dotsc,w_m\in W^a$ be data for a Schube…

Let $\gamma:\mathbb{R}\to\mathbb{R}_{m-1}[u]$ be a convex curve, meaning that any $m$ distinct points along $\gamma$ are linearly independent. Let $z_1,\dots,z_{d(m-d)}$ be distinc…

Let $\nu\vdash n$ and let $\boldsymbol{\mu}\Vdash\kappa$, where $\kappa=(\kappa_1,\dots,\kappa_s)$ is a composition of $n$. Let $I_1,\dots,I_s\subseteq\mathbb{R}$ be pairwise disjo…

Let $0\le k\le n$, and let $I_1,\dots,I_{k(n-k)}$ be pairwise disjoint intervals of $\mathbb{P}^1(\mathbb{R})$. For each $l$, let $X_l$ be a multiset of $k$ points in $I_l$, and le…

Let $V\in\operatorname{Gr}_{k,n}(\mathbb{R})$. Suppose that all complex zeros of the Wronskian $\operatorname{Wr}(V)$ are real, and let $I\subseteq\mathbb{R}$ be any interval on wh…

Consider a real Schubert problem for which Proposition … is sharp. The conjecture asserts that the cohomological lower bound can always be attained; the supplied excerpt gives no r…

Monotone Osculating Conjecture. For any such Schubert problem and any flags $F_{\bullet}^1,\ldots,F_{\bullet}^m$ osculating $\gamma$ at points that are monotone with respect to the…

Monotone Secant Conjecture. For any Grassmannian Schubert problem $(\sigma_1,\ldots, \sigma_m)$ on $\mathbb{F}\ell(\alpha;n)$ and any disjoint secant flags…

Let $G(k,n)$ be a Grassmannian, let $\gamma$ be a rational normal curve, and let $\lambda^1,\dotsc,\lambda^m$ be a Schubert problem on $G(k,n)$. A generalized secant subspace to…