8 problems

No periodic three-volume foam conjecture. There are no least-area divisions of $T^3$ into three volumes that lift to a foam in $\mathbb{R}^3$.

Let $\mathcal{C}_0$ be the decoration data for an $E_1$-monoidal $(\infty,n-1)$-category $\mathscr{F}_{n-1}(\mathcal{C}_0)$ of singular manifolds in $\mathbb{R}^n$, framed in codim…

Let $\mathrm{A}\boldsymbol{\mathrm{Link}}^+$ be the category of consistently oriented annular links and cobordisms, and let $K^b(\mathrm{A}\boldsymbol{\mathrm{Foam}}^+)$ be the hom…

Let $S$ be the surface supporting the tangled webs, let $\mathrm{S}\boldsymbol{\mathrm{Tan}}$ be the category of tangled webs and framed foams in four-dimensional space, and let…

Kronheimer–Mrowka's well-definedness conjecture. The quantity $J^\flat(F)$ is well-defined: it does not depend on the choices made in the seam-vertex cancellation step.

Let $\Sigma$ be a closed foam in $\R^{3}$, and let $S$ be its underlying closed pre-foam. Let $\Jsharp(\Sigma)$ denote the instanton-homological evaluation of $\Sigma$, and let…

Let $S$ be a closed pre-foam, namely an abstract finite $2$-dimensional cell complex with foam local structure, possibly with tetrahedral points and dots. Define $\Jflat(S)\in\F$ b…

Let $N\geq 4$ and consider the $\mathfrak{sl}(N)$-link homology defined in the preceding sections over the rational theory, with its relations and homomorphisms suggested to admit…