24 problems

Transitivity conjecture. The group spanned by mutations of exceptional bases and the isometries of $K_0(\mathbb{P}_n)$ acts transitively on the set of exceptional bases of…

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 $E$ 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…

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…

Contractibility conjecture. The principal connected component is the union of the geometric and algebraic stability conditions and is contractible:

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

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…

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 $n \in \mathbb{N}$ with $n \geq 2$, and let $\mu$ denote the $n$-dimensional Hausdorff measure on $\mathbb{S}^n$. Let $\mathcal{A} \subseteq \mathbb{S}^n$ be open and antipodal…

Let $X$ be a smooth projective variety, let $\mathscr{E}$ be an ample vector bundle on $X$, and let $T_X$ denote the tangent bundle of $X$. Andreatta–Wiśniewski's conjecture. If ……

Let $i$ and $n$ be such that $(i+1)\mid(n+1)$, and let the $i$-spaces of $\mathrm{PG}(n,q)$ be the relevant subspaces. Subspace-induced factorization conjecture. There exists a…

An $M$-cap in $\mathrm{PG}(n,q)$ is a set of $M$ points with no three collinear. By Ebert's theorem, if $n\geq 2$ is even, then $\mathrm{PG}(2n-1,q)$ can be partitioned into disjoi…

Let $n$ and $k$ be integers with $n>2k\geq 2$, and let $SG(n,k)$ denote the Schrijver graph. A quadrangulation of $P^{n-2k}$ is a graph embedded in the projective space $P^{n-2k}$…

Projection-map conjecture. These projection maps are homotopy equivalences. This conjecture concerns the topology of spaces of equivariant algebraic maps between real projective sp…

Let $A_d^{\mathbb{C}}(m,n)$ be the space of normalized $(n+1)$-tuples of homogeneous complex polynomials of degree $d$ defining algebraic maps from $\mathbb{R}\operatorname{P}^{m}$…

Let $m,n,d$ be integers with $2\leq m<n$ and $d\geq1$. Write $\operatorname{Alg}_d(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ and…