27 problems

Let $V$ be a $d$-dimensional subspace of $\mathbb{C}[u]$, and let the Schubert problem be defined by flags osculating the moment curve at real parameters. Shapiro–Shapiro conjectur…

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 $V$ be a $d$-dimensional vector space of real analytic functions on an interval $I\subseteq\mathbb{R}$, and let $\operatorname{Wr}(V)$ denote its Wronskian. Say that $V$ is dis…

Let $\rho_n:S^1\to\mathbb{R}P^n$ be the real rational normal curve, and let $\sharp_{k,n}$ denote the degree of the Grassmannian of projective $k$-dimensional subspaces in…

Let $X$ be an algebraic set defined over $\mathbb{R}$. It has the M-property when the sum of the $\mathbb{Z}/2\mathbb{Z}$-Betti numbers of $X(\mathbb{R})$ equals the corresponding…

Let $(w_1,\dotsc,w_m)$ be a Schubert problem for $\mathbb{F}\ell(\alpha;n)$, let $\Delta_w$ be its discriminant, and let monotonicity be defined using the descent sets…

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

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

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 $t,x_1,\dotsc,x_8$ be indeterminates, and define … and … Let $t_1<t_2<\dotsb<t_8$ be real numbers. Conjecture A. The polynomial system … … has $12$ solutions, and all of them a…

Let $a=(a_1<\dotsb<a_k)$ with $a_k<n$, and let $w_1,\dotsc,w_m$ be Schubert data for flags of type $a$. Let $F_\bullet^1,\dotsc,F_\bullet^m$ be osculating flags of the rational nor…

Let a zero-dimensional intersection of Schubert subvarieties of a Grassmannian be given by flags osculating a real rational normal curve at real points. The Shapiro conjecture. The…

Let $W_1,\ldots,W_q$ be projective subspaces in general position in $\mathbf{P}^d$, with $a_j={\mathrm{dim}\,} W_j$ satisfying … Let $E:\mathbf{P}^1\to\mathbf{P}^d$ be the rational…

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 $V\subseteq\mathbb{R}[u]$ be a finite-dimensional vector space of real polynomials, and let $\operatorname{Wr}(V)$ be its Wronskian. Let $I\subseteq\mathbb{R}$ be an interval a…

Let $V\in\mathrm{Gr}(d,m)$ be a point of a Grassmannian, and call it totally nonnegative if all Plücker coordinates are real and nonnegative up to rescaling. Call $V$ totally posit…

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 $\gamma$ be the moment curve and consider a Schubert problem in which each osculating plane $W_l$ to $\gamma$ is replaced by a plane spanned by $\gamma(x_1),\dots,\gamma(x_k)$,…

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…

Let $V$ be a complex vector space, let $\operatorname{Gr}(m,V)$ be its Grassmannian, and let $\operatorname{LG}(V)$ denote the relevant Lagrangian Grassmannian. Let…

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…