21 problems

Nonexistence conjecture. The polar space $\mathsf{Q}^+(9,q)$ does not have an ovoid.

Let $\mathcal{S}$ be a finite generalized quadrangle, and let $p$ be an odd prime. A central symmetry is a point symmetry of the generalized quadrangle, and the group of central sy…

Tits's center conjecture. Every convex subset $S$ of chambers in $\Delta$ must either contain a pair of opposite chambers, or $\mathrm{Stab}_G(S)$ fixes a non-empty partial flag of…

Thas's conjecture. J. A. Thas conjectured that the Segre construction for $q=3$ was the only example of a hemisystem on the Hermitian surface $H(3,q^{2})$.

Counting-function symmetry conjecture. For every $\kappa\in\mathbb{F}_q^*$, one has

Delsarte's conjecture. The linear programming bound for designs in association schemes should allow the non-existence results in Theorem to be extended from $\lambda=1$ to values s…

Coolsaet–De Beule–Siciliano conjecture. Every sharply transitive subset of $\operatorname{SL}(2,q)$ arises from a sharply transitive subgroup.

Let ${\mathcal P}$ be a rank $d$ polar space, and let $Y$ be a family of lines with characteristic vector $\chi_Y \in \langle \bm{j}\rangle + V_{10} + V_{11}$. Consider the example…

Non-regularity conjecture. The polar space associated to this induced form is non-regular.

Characteristic-free three-generators conjecture. The pair $\{M_1,M_2\}$ satisfies the three-generators property if and only if ${\cal S}$ admits a necessarily tight embedding

Embedding conjecture. Every regular polar space admits a tight embedding.

Let $\mathcal{P}$ be a finite classical polar space of rank $n$, and let a $t$-Steiner system be a collection of generators such that every totally isotropic $t$-space is contained…

Let $\Delta_i$ denote the graph associated with the $i$-dimensional subspaces of the finite polar space, and let $\kappa_i$, $\chi_i$, and $\xi_i$ be its corresponding graph parame…

Let $\mathbb{F}_0$ be a proper subfield of a field $\mathbb{F}$, and let $Q^+_{n-1}(2n-1,\mathbb{F})$ denote the polar Grassmannian of maximal singular subspaces of the hyperbolic…

Let $d$ be the dimension parameter of the polar space $\mathcal{P}$. A maximal chain in $\mathfrak{N}(\mathcal{P})$ is a chain of nondegenerate subspaces maximal under inclusion. W…

The discussion concerns pairs of point-line geometries associated with polar spaces whose point sets admit a common projective representation as a projective variety, while the lin…

Tight-set classification conjecture. Every $(q+1)$-tight set of $\mathcal H(4,q^2)$ that is not the union of $q+1$ lines is the set of points of an embedded $\mathcal W(3,q)$, or,…

Polar graph classification conjecture. Let $k\geq 2$. Then there exists $n_{q,k}$ such that every Boolean degree $1$ function $f$ on $C_q(n,k,e)$ with $n\geq n_{q,k}$ is trivial, t…

Let $Y$ be a maximum-size $(d,t)$-EKR set of generators in a polar space of rank $d$ and type parameter $e$, meaning that any two members of $Y$ meet in dimension at least $d-t$. A…

Dempwolff's conjecture. $n$-dimensional doubly dual hyperovals over $\mathbb{F}_2$ exist only if $n$ is odd.

Let ${\goth M}:=\VerSpace(k,\PolarSpace(\varpi))$, and let ${\mathscr H}$ be a hyperplane of \VerSpace(k,\text{\boldmathgoth P}) determined by a polarity $\varkappa$. The inter…