20 problems

A tiling of ${\mathbb R}^3$ into cells of unit volume is being considered, and the surface area of a cell is the quantity to minimize. The proposed minimizing cell is a slightly mo…

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…

For any link, let $\mathcal{F}(L)$ be the foam-based bigraded complex and let $\operatorname{KR}(L)$ be the KhovanovRozansky complex. Suppose the already established isomorphism be…

Consider any other evaluation of closed foams satisfying the same relations as the evaluation used in the thesis, and let it produce a functorial link homology categorifying the…

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 the Thom–Pontryagin construction produce a topological space representing cobordism classes of decorated foams, and let Quillen's K-theory space be the space representing the h…

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.

A foam is an equal volume equal pressure foam when its cells have equal volume and the same pressure. Kelvin's foam is the foam based on Kelvin's proposed equal-volume partition, w…

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…

Web-splitting tensor-product conjecture. The map

Let $w$ be an $\b5$-web, and let $f$ be a reduced $(w,w)$-foam. The identity foam proposition asserts that if $f$ is equivalent to $w\times I$ up to a non-trivial scalar, then $f$…

Consider clusters of four bubbles containing prescribed amounts of gas, and minimize the total area of the interfaces between the bubbles. A four-dimensional simplex is a polytope…

Let $k>1$ and let the KhovanovLauda 2-category categorifying quantum $\mathfrak{sl}_k$ be given. A foamation is a 2-functor from this 2-category to a 2-category of universal…

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…

Fix the foam relations used to define the foam-based link homology and the KapustinLi evaluation of closed foams. The relations are sufficient for the evaluation of closed foams wh…

For $N=3$, consider Khovanov's foam-based categorification of the $\mathfrak{sl}(3)$-link polynomial and the KhovanovRozansky categorification using matrix factorizations. Foammatr…

Let $N\geq 4$ and let the foam-based rational theory categorifying the $\mathfrak{sl}(N)$-link polynomial be the theory constructed in the thesis. A link homology is integral when…