292 problems

Braid-group fixed-point conjecture. For every $i\in[d-1]$, if $(k,n)$ is $(3,9)$ or $(4,8)$, every unstable fixed point of $\sigma_i$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$ is a c…

Shub's conjecture. There is a constant $c_{d,k}$ such that

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…

Integral cohomology and non-isomorphism conjecture. One has an isomorphism of integral cohomology groups

Let $\operatorname{Gr}(k,n)$ be a Grassmannian, let $(\lambda^1,\dotsc,\lambda^r)$ be a Schubert problem, and let $F^1_{\bullet},\dotsc,F^r_{\bullet}$ be flags secant to a rational…

Let $V(k,n)$ be the Zariski closure of the image of the rational Veronese map from $\operatorname{Gr}(2,n)$ to $\operatorname{Gr}(k,n)$, and let … denote its analytic closure. Lam'…

Grassmann convexity conjecture. The number of real zeros of this Wronskian in $I$ is at most

Let an oriented matroid be a combinatorial analogue of a real vector space, and let an OM-Grassmannian be the finite simplicial complex associated analogously to the Grassmannian o…

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 $\operatorname{Gr}(k,n)$ denote the Grassmannian of $k$-planes in $\mathbb{R}^n$, and let $\operatorname{Gr}_{\geq 0}(k,n)$ be its totally nonnegative part, consisting of the p…

Let $\Delta_{\lambda,\mu,\nu}$ be the triangular puzzle region whose left, right, and bottom boundary labels are respectively $\lambda$, $\mu$, and $\nu$. A KV-puzzle is a filling…

Let $D$ be a finite diagram contained in $[k] \times [n-k]$. Let $X_D$ be the associated diagram variety in $\operatorname{Gr}(k,n)$, with cohomology class…

Compatibly split positroid conjecture. The only compatibly split irreducible subvarieties of $\mathrm{Gr}(k,n)$ are the closed positroid varieties.

Williams's irreducibility conjecture. This set was conjectured to be irreducible. The claim is now a theorem, since the set equals the positroid variety…

Let $X=\operatorname{Gr}(k,n)$, let $\mathcal U$ be the tautological subbundle, and let $\mathcal A_i$ be the full triangulated subcategories defined in the source using the specif…

Mirror-family conjecture. The pair $M_y+W_y$ is the mirror family of $X_A+Y_A$, with $M_y$ and $W_y$ forming different parts of the B-model moduli space and $X_A$ and $Y_A$ forming…

Let ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of a building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$, and let ${\mathfrak G}_k$ be it…

Regularity conjecture. The CW-complex formed by the cells $S_\mathcal{M}^\mathrm{tnn}$ is regular. The closure of each cell is homeomorphic to a closed ball of appropriate dimensio…

Let $X=Gr_k(\mathbb{C}^n)$, and let $\sigma_{\lambda}$ and $\sigma_{\mu}$ be Schubert classes indexed by partitions contained in the $k\times(n-k)$ rectangle. Write … where…

Let $\mathbf{G}$ be the Grassmannian in its Plücker embedding, let $\mathbf{Pf}$ be its projective-dual Pfaffian variety, and write $\mathbf{P}^*$ for the dual projective space. A…

Covering property conjecture. For any $n>0$ and $1\leq k\leq n-1$, there exists a non-negative measure $\mu\in\mathcal{M}(G(n,n-k))$ supported in $Z$ such that

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…

The Schubert calculus conjecture. For generic $z=(z_1,\ldots,z_n)$, the intersection of Schubert varieties $\mathcal{I}_{\{{\bf w}\}}(z)$ is transversal. Here generic means that…

Let $F$ be a field, let $n$ be a positive integer, let $\mathcal{L}_n(F)$ be the degree $n$ component of the motivic Lie coalgebra, and let $\widetilde{\mathcal{L}}_n(F)$ be the gr…

Let ${{\bf G}}$ be the Grassmannian appearing in the source, let $\alpha$ be a cohomology class on ${{\bf G}}$, and let proposition … holds for arbitrary classes $\alpha$ on…