217 problems

Let $A=(a_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix, let $\beta\in Q_n^+$, and let $\Lambda\in P^+$ be a dominant integral weight. Write $\mathscr{R}_{\beta}^{\Lambda}$ fo…

Let $B_{(W,S)}$ be the generalized braid group associated with the Coxeter system $(W,S)$, and let $2$-${\mathcal B}r$ be the monoidal category generated by the Rouquier complexes…

Let $\mathcal{H} eis:= \mathcal{H} eis_{-1}(0)$ be Khovanov's Heisenberg category, and let its Karoubi envelope be its idempotent completion. Khovanov's categorification conjecture…

Crystal structure conjecture. The localization $\widetilde{\mathscr C}_w$ possesses a crystal $\mathbb B(\widetilde{\mathscr C}_w)$.

In the 1990s, Crane and Frenkel considered three-dimensional topological field theories of Witten, Reshetikhin and Turaev and the categorical language emerging in geometric represe…

Categorical braid group action conjecture. The functor $\cal{T}_{i,+1}'$ extends to an autoequivalence of $\textsf{Com}(\dot{\cal{U}}_Q(\mathfrak{g}))$ such that the induced automo…

Crane–Frenkel conjecture. There exists a categorification of the quantum group $U_q(\mathfrak{sl}_2)$ at roots of unity giving rise to a 4D TQFT.

Quilted Atiyah–Floer conjecture. The three extended Floer field theories induce isomorphic $2$-functors

Functoriality conjecture. This assignment of a map to a cobordism is independent of the choice of Morse function, i.e. this makes the knot homology theory $\mathcal{K}(-)$ functori…

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…

Let $D$ be a plane diagram of a link, let $C(D)$ be the associated complex of graded $\mathbb Z[c]$-modules, and let quasi-isomorphisms between such complexes be associated to Reid…

Let $D$ be a generic plane projection of an oriented link $L$ in $\mathbb R^3$, and let ${\cal H}^{i,j}(D)$ be the finitely generated bigraded groups constructed from $D$, whose is…

Let $V_{\lambda}=\bigoplus_{\mu}V_{\lambda}(\mu)$ be a finite-dimensional irreducible representation of $\mathfrak{sl}_m$, categorified by blocks of Ariki–Koike cyclotomic Hecke al…

Let $k>1$ and let $L$ be an oriented $k$-component link. Let $\widetilde{\mathrm{H}}(L)$ denote the proposed redefined triply graded link homology theory. Finite-dimensional extens…

Let $W$ be the Weyl group, let $S_{\bf d}$ be the subgroup associated with the tensor-product datum ${\bf d}$, and let $H_{\bf d}^i=(B^{\bf d}\otimes C)^{W_i}$ and…

Let $V_1$ be the two-dimensional fundamental representation of $obreakfrak{sl}2$, and consider the two categorifications of the $Uq(mathfrak{sl}2)$-action on $V1^{otimes n}$ const…

Let $\mathcal A$ be the category of $\mathbb Z[1/2]$-modules. Let ${\bf\mathcal Cob}_f^4$ be the category of colored framed link diagrams and movie presentations of colored framed…

Categorification conjecture. The object $\Theta(M)$ should have at least the properties specified subsequently in the source, categorifying the normalized WRT invariant…

Decategorification conjecture. The decategorification of ${\mathcal{B}}_W$ is equal to $B_W$.

Let $D$ be a diagram of an oriented framed link $L$ whose components or strands are colored by numbers from $1$ to $n-1$, and let $P_n(L)$ be the corresponding colored link invaria…

Let $\Gamma$ be a labeled trivalent graph, let $H(\Gamma)$ be the cohomology of its associated matrix-factorization complex, and give the coefficient rings $S(e)$ the grading deter…

Let $H_3$ denote the doubly graded homology theory constructed from the matrix-factorization complexes in the paper, and let the homology theory constructed in [Kh3] be tensored wi…

Let $R$ be a finite-dimensional unital based ring, and let ${\mathcal C}$ be a rigid monoidal category whose Grothendieck ring is isomorphic to $R$. Ocneanu rigidity. For a fixed…

Parabolic Temperley–Lieb relations conjecture. These functors should satisfy the natural isomorphisms corresponding to the defining relations of the Temperley–Lieb category, with…

Let $\mathcal O^{k,n-k}$ be the relevant parabolic category, and let $\mathcal U_i$ for $1\leq i\leq n-1$ be the projective functors defined by $\mathcal U_i=T_i\circ T^i$. Product…