19 problems

Let $C_5$ be the $5$-cycle and let $K_n$ be the complete graph on $n$ vertices. The 5-cycle complex $\operatorname{Hom}(C_5,K_n)$ is the Lovász complex of graph multimorphisms from…

Let $V_{n,m}$ denote the Stiefel manifold of orthonormal $m$-frames in ${\mathbb R}^n$, and let the higher-rank analogue of the cosine transform be the cosine transform on…

Let $C_5$ be the cycle graph on five vertices, let $K_{n+2}$ be the complete graph on $n+2$ vertices, and let $\operatorname{Hom}(C_5,K_{n+2})$ be its graph homomorphism complex. L…

Injectivity conjecture. The projected normal random variable has inherent average $\mathbb{E}[X]$. Moreover, there exists an injective function

Let $\mathbb{K}$ be the underlying scalar field, let $\operatorname{M}_{n,k}(\mathbb{K})$ be the ambient matrix space, and let $\operatorname{St}_{n,k}(\mathbb{K})$ be the Stiefel…

Injectivity-radius conjecture. The displayed equalities hold for the respective ranges of $\beta$. The result gives only an interval estimate when $\beta\in(\frac13,\frac23)$ and o…

Let $\mathrm{St}(n,p)$ be the Stiefel manifold, let $U,\widetilde{U}\in\mathrm{St}(n,p)$, and let $U_N,\widetilde{U}_N\in\mathrm{St}(n+N,p)$ be obtained by appending $N$ zero rows…

Let $n$ and $p$ be integers with $p\leq n$, let $\beta>0$ parametrize the Riemannian metric on the Stiefel manifold $\mathrm{St}(n,p)$, and consider the lower bound for the corresp…

Absil et al.'s injectivity-radius conjecture. For the canonical metric on the Stiefel manifold, the injectivity radius is

Let $\mathrm{St}(n,p)$ be the Stiefel manifold with $2\leq p\leq n-2$, equipped with a metric from the one-parameter family indexed by $\beta>0$. Let $t^{\mathrm{r}}_\beta$ be the…

The complex Stiefel manifold $\mathrm{St}(r,p,\mathbf{C})$ is simply connected as soon as $p\geq r+1$. The optimality conjecture. This condition is correct, so the sufficient condi…

Let $S$ be a closed surface, let $\mathrm{Hit}_{2n}(S)$ and $\mathrm{QHit}_{2n}(S)$ denote the Hitchin and quasi-Hitchin representation spaces, respectively, and consider the geome…

For $n$ in the range considered by the complex Stiefel manifolds, let $V_2\mathbb{C}^{n+2}=\operatorname{SU}(n+2)/\operatorname{SU}(n)$ be the complex Stiefel manifold of orthonorm…

Almost global synchronization conjecture. The consensus manifold $\mathcal{C}$ is almost globally asymptotically stable for all such $p$; equivalently, $p\leq n-2$ is both necessar…

Sharp inequality conjecture. A sharp inequality for almost-global asymptotic stability is

Upper kissing-radius conjecture. The upper bound $\bar{\varrho}$ can be generalized to all Stiefel manifolds. This would extend the stated unitary-group result beyond the unitary g…

Pointwise inversion conjecture. (a) For every $v\in V_{n,m}$, $(\mathscr{M}^\alpha f)(v)$ extends meromorphically in $\alpha$ with polar set ${\mathbb N}=\{1,2,3,\ldots\}$. (b) For…

Let $V_{n,m}$ be the Stiefel manifold, let $f$ be an integrable function on $V_{n,m}$, and let $\omega\in C^\infty(V_{n,m})$. Define … where $Q^\alpha$ is the analytically continue…

Let $f\in C^\infty(V_{n,m})$, and let $T^\alpha_{m,k}$ denote the cosine or sine transform under consideration. For $\omega\in C^\infty(V_{n,k})$, define the distribution-valued me…