94 problems

Let $f_n$ be the chain-type invertible polynomial, let $S_n$ and $L_{f_n}$ be the associated ring and grading group, and let $\omega_1,\dots,\omega_n$ be positive rational numbers…

Berglund–Hübsch homological mirror symmetry conjecture. There exists a derived equivalence between $\mathcal{F}(W,G)$ and $\mathcal{MF}(W^T,G^T)$.

Let $Q$ be a regular local ring of dimension $n+1$ with algebraically closed residue field, and let $e$ denote the floor of $\frac{n}{2}$, which we write as…

Let $\mathbf{w}$ be an invertible polynomial in $n$ variables, let $\Gamma_{\mathbf{w}}$ be its maximal grading group, and let $\widecheck{\mathbf{w}}$ be its Berglund–Hübsch trans…

Boundary converse conjecture. If a positive matrix $M \in \mathcal{M}_{\binom{k+1}{2},k}^{p \times q}$ does not have a size-$k$ psd factorization in which all factors have rank $k$…

Let $\mathbf{w}$ be an invertible polynomial with admissible group of symmetries $\Gamma$, and let $\widecheck{\mathbf{w}}$ be its Berglund–Hübsch transpose with dual grading group…

Let $A=(a_{ij})$ be an $(n+1)\times(n+1)$ integer matrix with nonzero determinant, and define … Let $\check{{\bf w}}$ be the polynomial associated to $A^T$, let $\Gamma_{\bf w}$ be…

Bézout-domain conjecture. If $R$ satisfies property $\mathrm{ID}_2$, then $R$ is a Bézout domain.

Let $f\in S^n$ be an invertible polynomial, and let $\widetilde{f}$ denote its Berglund–Hübsch transpose. Let $L_f$ be the maximal grading of $f$, and let…

Let $\mathcal{X}$ be the versal deformation over $\operatorname{PRes}$ of a partial resolution of a rational double point corresponding to a single Dynkin-diagram vertex whose coef…

Let $X$ be the hypersurface of a flop of length $\ell$, and let $M$ and $M^+$ be the maximal Cohen--Macaulay modules associated with the two resolutions. A matrix factorization of…

Let $X$ be the contracted threefold of a flop of length $\ell$, with resolutions $Y$ and $Y^+$, and let a maximal Cohen--Macaulay module on $X$ have rank $\ell$. The Grassmann blow…

A rational double point is a surface singularity of type $A_{n-1}$ or $D_n$, and a partial resolution dominated by its minimal resolution has an associated maximal Cohen--Macaulay…

Let $f$ be the defining polynomial of a simple singularity, and let $D^b_{\mathbb Z}(\mathcal A_f)$ denote the triangulated category of graded matrix factorizations of $f$. Let $Q$…

Let $W$ be a regular weight system corresponding to an ADE singularity, and let $f$ be a quasi-homogeneous polynomial attached to $W$. Write $D^b_{\mathbb Z}({\mathcal A}_f)$ for t…

Let $W$ be a regular weight system and $f$ be a quasi-homogeneous polynomial attached to $W$. Assume $W$ has a dual regular weight system $W^*=(a^*,b^*,c^*;h)$ in the sense of K. S…

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 $\Gamma$ be a labeled trivalent graph, let $C(\Gamma)$ be the associated tensor-product $2$-complex of matrix factorizations, and let $H(\Gamma)$ be its cohomology. When loops…

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 $G$ be the group acting in the above setup, and let $a_i$ and $x_i$ be the coefficients and coordinates defining the quadratic potential … Here $\operatorname{MF}$ denotes the…

Let $\mathcal{M}^{\mathrm{vert}}(\mathbb{P})$ be the vertical moduli space of a good degeneration, equipped with the global $(-1)$-shifted potential function $\mathbb{W}=f\cdot g$.…

Let $U$ be a smooth $\mathbb{C}$-scheme and let $\phi\colon U\to\mathbb{C}$ be a regular function. Let $\mathbf{M}\to\mathbf{Crit}(\phi)$ be a $(-1)$-shifted Lagrangian such that…

Let $\mathrm{X}_{1,i}$ be a general hourglass, let $\widetilde{\mathrm{X}}_{1,i}$ be its associated space, and let $L_{i_j,\mathrm{p}_j}$ be any stacky line through the $j$-th orbi…

Let $\mathrm{X}_{1.9}$ and $\mathrm{X}'_{1.9}$ be the hourglass varieties, and for each $i$ let $\mathcal{MF}_{i}(\mathrm{X}_{1.9})$ denote the left orthogonal component of the exc…

Cohomological conjecture. For each combinatorially spin Delzant polytope $\triangle$ such that $X_\triangle^\mathbb{R}$ is Fano, we have