51 problems

Matsuki–Gindikin conjecture. For every $K_\mathbb C$-orbit $S$ of nonholomorphic type on $X$, one has

Finite-part factorization conjecture. There are positive integers $d_1,d_2,\ldots,d_l$, with $l\leq n$, such that

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…

Homotopy-generator conjecture.

Polynomiality conjecture. $\bar Q_A(q)$ and $\bar R_A(q)$ are polynomials in $q$, and they correspond to the finite part and the infinite polynomial part of $H^*(F(A))$.

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…

Casian–Kodama conjecture. The graph ${\mathcal G}$ is the graph of incidence numbers for the integral cohomology of the real flag manifold ${\mathcal B}$ ($\hat{\mathcal B}$) in te…

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 $G/T$ be a flag manifold, let $F(n)$ be the complete flag manifold, and let $Map(G/T,F(n))$ denote the space of continuous maps $G/T\to F(n)$. For $f\in Map(G/T,F(n))$, 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 $t,x_1,\dotsc,x_8$ 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 $F(n)$ be a full flag manifold, and consider families of (1,2)-symplectic metrics on $F(n)$ that are different from the family of Kähler metrics. Dimension conjecture. All such…

Let $F(n)$ be a full flag manifold with $n\geq 5$, equipped with a complex structure $J$ and its associated tournament ${\cal T}(J)$. Let $ds^2_{\Lambda}$ denote the Borel metric.…

Kahler volume formula. The Kahler volume of the space ${\cal M}_d$ of parametrized rational curves of degree $d=(d_1,\dots,d_n)$ with respect to the Kahler form with periods…

Let $\mathcal{G}_{\kappa_1,\dots,\kappa_{k+1}}^m$ be the Hermitian Jacobi operator on the simplex, and let the multivariate Schur polynomials be the basis of unitary-conjugation-in…

Let $m$ be a positive integer, let $k$ be the number of matrix variables, let $(\kappa_1,\dots,\kappa_{k+1})$ be the parameters of the Hermitian Jacobi operator, and let…

Podestà–Zuddas conjecture. Every $G$-invariant BTP metric on $(G/H,J)$ is either a Kähler metric or a multiple of the standard metric $g_o$.

Podestà–Zuddas conjecture. Every $G$-invariant BTP metric on $G/H$ is either a Kähler metric or a multiple of the standard metric.

Let $G$ be a simple Lie group and let $P$ be a parabolic subgroup of $G$. A domain in $G/P$ is proper almost-homogeneous if it is proper and admits an automorphism group with a com…

Let $G/P$ be a flag manifold, where $G/P$ is neither real projective space nor the conformal sphere. A proper divisible domain is a proper domain in $G/P$ admitting a properly disc…

Let $M=\mathsf G/\mathsf H$ be one of the low-dimensional full flag manifolds listed in Theorem, with $\mathsf G$ a compact simple Lie group of type ${\rm A}_n$, ${\rm B}_n$,…

Let $M$ be a flag manifold with an almost-complex structure $J$ and a quasi-Kähler metric $\lambda$ whose curvature is Griffiths semi-positive. Let $(M_i,g_i)$ denote Hermitian sym…