26 problems

Let $(Q,\mathcal{S})$ be a Jacobi-finite connected quiver with potential. A decorated representation is rigid when it has no nontrivial self-extension, a negative representation is…

Let $B$ be a skew-symmetrizable integer matrix, let $\mathcal{A}(B)$ be the associated cluster algebra with the particular choice of coefficients specified in the source, and let…

Mutant subpopulation size conjecture. The actual size of the mutant subpopulation at time $t(v)$ is of order

Floer-rank mutation conjecture. The ranks of $I^\#$ and $\widehat{\mathit{HF}}$ are preserved by genus-2 mutation.

Genus-2 mutation invariance conjecture. The values of $\tau^\sharp$ and $\tau$ are preserved by genus-2 mutation.

Let $\operatorname{\mathcal{D}}$ be a link diagram, let $p$ and $p'$ be points that are not double points of $\operatorname{\mathcal{D}}$, and let…

Gillespie's neutrality conjecture. In this regime, the behavior should look like that of a neutral model.

Conjecture A. There exists a bijective correspondence between the set of mutation-equivalence classes of Fano polygons and the set of qG-deformation equivalence classes of locally…

Let $L$ and $L'$ be mutant links, and let $T$ be a tangle satisfying the hypotheses of the paper's mutation-invariance theorem for delta-graded knot Floer homology. Positive-mutati…

Period-determines-mutation conjecture. If $P_1$ and $P_2$ have the same singularity content and there is an affine-linear isomorphism $\varphi\colon L_{P_1}^T\to L_{P_2}^T$ such th…

Conjecture A. There is a one-to-one correspondence between $\mathfrak{P}$ and $\mathfrak{F}$. The correspondence sends $P$ to a generic qG-deformation of the toric surface $X_P$.

Degree and ramification conjecture. The following hold:

Work in the setting of Cohen–Macaulay rational singularities. Let $N=H^0(\mathcal{V}_X)$, where $\mathcal{V}_X$ is the object from the paper’s contraction and tilting setup, and le…

Let $I$ be the index set and let $M$ be the module in the moduli-tracking setup. Let $\upnu_I$ denote mutation at $I$, and assume $\upnu_I\upnu_I M\cong M$. Moduli tracking conject…

Let $C_i$ be a crepant curve on a $3$-fold $X$, and suppose that its universal sheaf is perfect and that $X$ has at worst Cohen–Macaulay rational singularities. Mutation theorem fo…

Source-sink denominator recursion conjecture.

Property R conjecture. Property R holds when mutation at $k$ is a source-sink move.

Source-sink recursion conjecture. The initial-seed-mutation recursion holds in the case of source-sink moves in arbitrary cluster algebras.

Let $K$ and $K'$ be knots related by Conway mutation, and let the $\delta$-grading on $\widehat{HFK}$ be the difference between the Alexander and homological gradings. Szabó's conj…

Let $(\Sigma,\mathbb{M})$ be a positive-genus surface with empty boundary and exactly two punctures, and let $\tau$ be a tagged triangulation of $(\Sigma,\mathbb{M})$. Write…

Let $\mathbb{T}_n$ be the $n$-regular tree indexing seeds, let $t_0\stackrel{k}{\text{---\!---}}t_1$ be an edge, and let $B_0$ and $B_1$ be exchange matrices with $B_1=\mu_k(B_0)$.…

Admissibility preservation conjecture. Admissibility is preserved in the mutation class of any acyclic diagram.

Mutation-admissibility conjecture. The mutated matrix $A'$ is also admissible.

Quasi-Cartan companion conjecture. Any diagram in the mutation class of an acyclic diagram has an admissible quasi-Cartan companion which is equivalent to a generalized Cartan matr…

Let $(D',P)=\tau(D,P)$ be obtained from a connected link diagram with basepoint by component-preserving Conway mutation. Let $\widetilde{C}(D)$ denote the reduced Khovanov complex,…