18 problems

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 Schubert problem for $\mathbb{F}\ell(\alpha;n)$. For each permutation $w_i$, 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 $\mathbb{F}\ell(\alpha;n)$. 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…

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

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.…

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 $g_o$.

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 $M$ 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…

Generalized singularity conjecture. If $\mathsf{m}$ has finite support, then the $\mathsf{m}$-stationary measure $\nu$ is singular to every coarse Iwasawa Patterson–Sullivan measur…

Let $\operatorname{Conf}_n(\mathbb{R}^3)$ be the configuration space of $n$ distinct points in $\mathbb{R}^3$. For a configuration $(p_1,\dots,p_n)$, let $\ell_i(p_1,\dots,p_n)$ be…

Let $\operatorname{Fl}_{\mathcal{D}}$ be a real flag manifold, and let $H^*(\operatorname{Fl}_{\mathcal{D}};\mathbb{Z})$ denote its integral cohomology. More generally, let an $R$-…

Let $\Delta$ be a flag 3-manifold, and let $\beta_1(\Delta)$ denote its first Betti number. 3-dimensional manifold Charney–Davis conjecture. … The conjecture extends the Davis–Okun…

Let $G$ be a connected non-compact simple Lie group with trivial center, and let $P \leq G$ be a parabolic subgroup. A convex divisible domain in $G/P$ is a bounded open convex sub…