199 problems

Kontsevich's homological mirror symmetry conjecture. Under the mirror map $\Greekmath 0124 \longleftrightarrow\Omega^{\vee}$ and $\Omega\longleftrightarrow\Greekmath 0124 ^{\vee}$,…

Fukaya-category elliptic-curve summand conjecture. The Fukaya category of $X$ should contain a direct summand equivalent to the derived category of sheaves on a genus-one curve who…

Let $(X,\omega)$ be a symplectic manifold equipped with a Lagrangian torus fibration admitting a Lagrangian section. Kontsevich–Soibelman associate to this data a rigid analytic sp…

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

Hochschild cohomology quotient conjecture. There exists a $\mathbb{C}[s]$-module structure on $HH^*(\mathcal{W}(X))$ such that $HH^*(\mathcal{W}(\underline{X}))$ is isomorphic to t…

Open string mirror symmetry conjecture. Under mirror symmetry, there is a quasi-equivalence of triangulated categories

Joyce's conjecture. The holomorphic volume form $\Omega$ defines a stability condition in the Fukaya category of $X$.

Let $X$ be a symplectic manifold, let $\mathcal{F}uk(X)$ denote its Fukaya category, and let the Gromov–Witten invariants of $X$ be its symplectic enumerative invariants. Kontsevic…

Joyce's conjecture. There exists a Bridgeland stability condition $\sigma=(Z,\mathcal{P})$ on $\operatorname{DFuk}(M)$ such that the central charge is induced by

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…

Singular complete-intersection Knörrer periodicity conjecture. Under appropriate hypotheses on $f_1,\dots,f_k$, there is a quasiequivalence of $A_\infty$-categories

Let $(M,\omega)$ be a symplectic manifold with $c_1(M)=0$, let $(X,J)$ be a complex manifold forming a hypothetical mirror pair with $(M,\omega)$, and let…

Let a complex structure determine a Fukaya category, and let special Lagrangians be the relevant geometric objects in that category. Stability conjecture. Complex structures are st…

Let $\mathbb{T}_0$ be the punctured torus, and let $D^\pi\mathcal{F}(\mathbb{T}_0)$ denote the split-closed derived Fukaya category of $\mathbb{T}_0$. Let the constructible plumbin…

Hypertoric Fukaya-category equivalence conjecture. There is a natural equivalence between

Let $J_1$ and $J_2$ be choices of almost complex structure, and let ${\Bbb R}_1$ and ${\Bbb R}_2$ be the corresponding elements of the completed exterior-algebra construction. Let…

Let $R^k(\alpha)=\sum_{g=0}^{\infty}R^k_g\alpha^g$ be the formal power series whose coefficients are defined by counts of holomorphic maps from a genus-$g$ curve with $k$ boundary…

Let $\Delta$ be a convex lattice polygon containing the origin in its interior. A Laurent polynomial $W$ has Newton polygon $\Delta$, and a pair $(G,D)$ consists of a consistent di…

Fix a field $k$ and positive integers $p_0,\ldots,p_n>1$. For each $i$, let $W_{p_i}(X_i)\in\mathbb{C}[X_i]$ be a general polynomial of degree $p_i$, and set … Let…

Let $(M,\omega)$ be an integral symplectic manifold, meaning $[\omega]\in H^2(M;\mathbb Z)$. Let $\mathfrak h$ be the upper half-plane, set $M_\tau=(M,-\sqrt{-1}\tau\omega)$ for…

Fukaya-category Hochschild–Floer isomorphism conjecture. In good circumstances, this map is an isomorphism. This would identify the Hochschild homology of the Fukaya category with…

Let $Y_1$ and $Y_2$ be manifolds with mirror Landau–Ginzburg models $(X_1,W_1)$ and $(X_2,W_2)$. The mirror of $Y_1\times Y_2$ is $(X_1\times X_2,W_1+W_2)$, and let…

Let $\mathcal F(X)$ be the Fukaya $A_\infty$-category of a Calabi–Yau manifold $X$, and let $\mathbf D\mathcal F_0(X)$ be its derived Fukaya category. Let $QH^*(X)$ denote the quan…

Let $M$ be an exact symplectic manifold embedded in a closed symplectic manifold $X$ as in Assumption. Let $\mathcal F(M \subset X)_{gen}$ be the generic fibre of the $A_\infty$-de…

Let $M$ be an exact symplectic manifold satisfying Assumption, and let … be the natural open–closed string map from symplectic cohomology to the Hochschild cohomology of the Fukaya…