41 problems

Let $\operatorname{\mathbb{R}P}^n$ denote real projective $n$-space, and let its length mean the invariant defined in the paper. Length monotonicity conjecture. The length of … is…

Let $N$ be the distinguished element of the collection, and let $\mathcal{S}$ be a collection of non-overlapping subsets of $N$. For $S,T\in\mathcal{S}$, write $S\prec T$ when…

Let $E$ be a rank $r\geq 2n$ locally-free instanton sheaf of trivial splitting type on $\mathbb{P}^{2n}$ and charge $c$. Lifting conjecture. There is a rank $r$ locally-free instan…

Let $N\geq 2$, let $1\leq k\leq 2N-2$, set $d=k+1$, and let $\mu>0$. For the Berger metric $g_\mu$ on $\mathbb {RP}^{2N-1}$, define the mod-two homological systole by … For a real…

Non-rigidity conjecture. For every $n\geqslant 4$ and every finite subgroup $G\subset\mathrm{PGL}_{n+1}(\mathbb{R})$, $\mathbb{P}^n_{\mathbb{R}}$ is not $G$-birationally rigid.

Let $\mathrm{SP}^d(\mathbb{P}^n)$ be the $d$-th symmetric product of complex projective space, and let $cl$ denote the cycle class map from Lawson homology to singular homology wit…

Let $\chi_2(n)$ denote the chromatic number of the binary projective space $\mathrm{PG}(n-1,2)$. Thus $\chi_2(4)=3$, $\chi_2(5)=4$, and $\chi_2(6)=5$. Haddad's conjecture. The chro…

Let $\Omega=\mathbb{FP}^d$ be a projective space, and let $I_{F_s}$ denote the energy associated with the kernel $F_s$. Let $s^*\in[-2,-1]$ be the unique value such that $I_{F_s}$…

Integral Fejes Tóth conjecture. The maximum of $I_1(\mu)$ over all Borel probability measures on $\mathbb S^d$ is achieved by $\mu_{ONB}$. The discrete conjecture remains open for…

Let $\mathbb{P}^n$ be complex projective space, and let $UH(\mathbb{P}^n)$ denote the largest integer such that every uniform vector bundle of rank at most that integer on…

A strong $(t\mod q)$-arc in a projective space is a point set with the corresponding arc multiplicity condition; an arc is lifted if it is obtained by the lifting construction from…

Let ${\operatorname{PG}}(n,q)$ be the finite projective space of dimension $n$ over the field with $q$ elements, and let $\chi'({\operatorname{PG}}(n,q))$ denote its chromatic inde…

Let $L\subset\mathbb{C}P^n$ be a Lagrangian submanifold satisfying $2H_1(L;\mathbb{Z})=0$. The source notes that such an $L$ has minimal Maslov number $n+1$ and is homotopy equival…

The projective-space characterization conjecture. Then $X\cong \mathbb{P}^n$.

Motivic vector bundle conjecture. The answer to the question whether every topological vector bundle is motivic should always be yes. In particular, on $\mathbb{P}^n$, every topolo…

Let $X$ be a Hausdorff topological space, and let $(X_n)_{n\in\omega}$ be a superskeleton of $X$. For every $n<m$ in $\omega$, consider the subset $X_m$ of $X_n$ and the complement…

Projective-space stability conjecture. There exists a sufficiently small $\varepsilon>0$ such that, for every $0<\delta<\varepsilon$, both…

Let $m\geq 2$, and let $F$ and $G$ denote the Fréchet functions associated with the intrinsic and comparison distance settings, respectively, near the origin $0$ in the real projec…

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, and let $I_{\mathbb{P}^{n+1}(\mathbb{K})}$ denote the isoperimetric profile of $\mathbb{P}^{n+1}(\mathbb{K})$, defined by ……

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, and let $\mathbb{P}^{n+1}(\mathbb{K})$ be the corresponding projective space. A projective subspace…

Ma's conjecture. Theorem 4 should hold in the stated generality: the Euclidean isotropic covariance matrix function and the corresponding covariance matrix functions obtained using…

Let $\operatorname{D}^b(\operatorname{Coh}\mathbb{P}^2)$ be the bounded derived category of coherent sheaves on the projective plane, and let…

Let $n\geq 1$, and let $\operatorname{D}^b(\operatorname{Coh}\mathbb{P}^n)$ be the bounded derived category of coherent sheaves on projective $n$-space. Let…

Let $d4d5$ be a codimension one foliation on $d4d9^n$, with $n d45e 3$. An invariant algebraic hypersurface is an algebraic hypersurface preserved by $d4d5$; $d4d5$ is everywhere t…

Let $\mathrm{PG}(N,q)$ be a projective space, and let $U_w$ denote the number of uncovered points after a $w$-cap $\mathcal{K}_w$ has been selected in the iterative process. Let…