13 problems

Rubin's saturation conjecture. Cyclic groups are saturated as long as the prime factors of their order are sufficiently large.

Shortest-path bisaturation conjecture. A shortest path between the trivial $G$-transfer system and the complete $G$-transfer system on a Hasse diagram will pass through as many bis…

Blumberg–Hill's classification conjecture. $N_\infty$-operads are completely classified by their associated $G$-transfer systems.

Disklike characterization conjecture. The maximal compatible transfer system is

Non-abelian generalization conjecture. The assumption that $G$ is abelian can be removed from the theorems identifying $\mathrm{Hull}(\mathcal{D}(\mathcal{U}))$ with…

Realization conjecture for compatible pairs. Every compatible pair of transfer systems can be suitably realised as an operad pair.

Abelianness conjecture for saturated groups. If $G$ is a saturated finite group, then $G$ is abelian.

Two-component LSP compatibility conjecture. Then $T$ is not LSP and is compatible with the transfer system generated by adding the edge $e\to G$ to $T$.

Let $G$ be a finite group. A pair of $G$-transfer systems is called compatible when it satisfies the compatibility conditions for a pairing of $N_{\infty}$-operads, and it is reali…

Let $n\geqslant 0$. A transfer system in $T(n)$ is maximally extendable if it has the maximal extension property described in the surrounding definitions, and let $\mathfrak{A}_m$…

Let $G = \operatorname{SL}_2(\mathbb{F}_p)$ with $p > 5$, $p \neq 11$, and $p \equiv \pm 3 \mod 8$. Fix an arbitrary transfer system $\mathcal{R}$ on $G$, and let…

Fix positive integers $r_1,\ldots,r_k$, and let $p_1,\ldots,p_k$ be distinct primes. A saturated transfer system on a finite group is a transfer system satisfying the two-out-of-th…

A $G$-indexing system is a collection of finite $H$-sets for subgroups $H\leq G$ satisfying the indexing-system axioms; these collections form a poset $\mathcal{I}(G)$ under inclus…