14 problems

Van den Bergh's conjecture. Any three-dimensional Calabi–Yau algebra derives from a potential.

Let $\Lambda=kQ/I$ be a finite-dimensional algebra whose quiver $Q$ is strongly connected, and let $S_x$ be the simple module associated with a vertex $x$. Write $\Omega^n(S_x)$ fo…

Proper containment conjecture. The ideal of polynomial identities of the path algebra is properly contained in that of its principal subalgebra:

Let $\Lambda=KQ/I$ be a TSP4 algebra with biregular Gabriel quiver $Q$. A vertex of $Q$ is a 1-vertex if it has the corresponding local configuration in the Gabriel quiver; the pre…

Twisted Calabi–Yau conjecture. $H(\alpha,\beta,\gamma)$ is twisted Calabi–Yau in general.

Strong no-loop conjecture. The simple module $k_x$ associated to the vertex of the loop has infinite projective dimension:

Toric-CY-four-fold magnificent-four conjecture. After suitable conditions, the deformation parameters reduce to at most three independent parameters; the D0/D8 operators are the st…

Let $(Q,W)$ be a quiver with potential, and let $J_{(Q,W)}$ be its Jacobian algebra. A finite-codimensional ideal is an ideal of finite codimension in this algebra. Ideal-counting…

Consider quiver BPS algebras associated with toric Calabi–Yau threefolds, and suppose that a higgsing operation relates a parent theory to a higgsed theory. In the rational and tor…

Let a toric Calabi–Yau threefold be described by a quiver, and let the quiver determine its associated quiver BPS algebra. Seiberg or toric duality produces a dual quiver and its a…

Weaker puncture hypothesis conjecture. Theorem Main2 holds under the weaker hypothesis

Let $\Gamma$ be a directed layered graph, and let $A(\Gamma)$ be the algebra associated with $\Gamma$. Koszulity conjecture for $A(\Gamma)$. Every algebra $A(\Gamma)$ is Koszul. Th…

Noncommutative resolution conjecture. At least in physically relevant examples, this space should be a noncommutative resolution of the singular center $Z$, because D-branes suppos…

R-charge volume conjecture. The $R$-charge of the arrow $a$ is the volume of this zero locus in $\operatorname{Max}Z$. In physics terms, the $R$-charge of a field is the volume of…