103 problems

In the appropriate Fukaya-category setting for a compact Calabi–Yau manifold $X$, if a Lagrangian submanifold $L\subset X$ defines an object $E_L$ that is stable with respect to th…

Let $\Sigma_g$ be a closed oriented surface of genus $g\ge 2$, let $\Lambda$ be the complex Novikov field, and let $\mathscr F_g$ be the split-closed, strictly unobstructed, two-pe…

Let $W$ be an invertible weighted homogeneous polynomial in $n+1$ variables, let $\check{W}$ be its Berglund–Hübsch transpose polynomial, and let $\Gamma_W$ be the group acting dia…

Let $X$ and $Y$ be mirror varieties, let $\mathcal{F}uk(X)$ denote the Fukaya category of $X$, and let $D^b\operatorname{Coh}(Y)$ denote the bounded derived category of coherent sh…

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…

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…

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

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…

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…

Sheaf quantization and Fukaya category conjectures. For each Lagrangian brane $\bm L$ in $X$, there is an almost quasi-isomorphism

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be an $A_\infty$-functor, and let $HC^-_*(F)$ be its relative cyclic homology equipped with the relative connection $\nabla^F$. Relative c…

Let $X$ be a symplectic manifold and let $L\subset X$ be a null-homologous Lagrangian. Let $HC^-_*(Fuk(X),L)$ denote the relative cyclic homology associated to the object $L$, and…

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…

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 $M$ 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…

Let $c,c'$ be matching paths, let $\Lambda_c$ and $\Lambda_{c'}$ be the associated eventually Lagrangian submanifolds, and let $\mathcal A,\mathcal A'$ be directed $A_\infty$-categ…

Let $(E,\pi)$ be an exact Morse fibration over $D$ with relative Maslov map $\delta_{E/D}$, let $M$ be the fibre over the base point, and let $\mu\in \operatorname{Symp}^e(M)$ be t…

Let $X$ be a symplectic manifold, let $F^{naive}(X,\omega)$ denote its naive Fukaya category, and let $D^b_{\infty}(hol(X))$ be the derived $A_{\infty}$-category of holonomic modul…

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…

Let $\mathfrak{L}$ be a Solomon–Verbitsky collection of Lagrangian submanifolds in a holomorphic symplectic manifold $(\mathrm{X},\sigma)$. Let $\widehat{\mathcal{A}}_\mathfrak{L}$…

Let $(\mathrm{X},\mathrm{I},\mathrm{J},\mathrm{K},g)$ be a hyperkähler manifold, and let $\mathfrak{L}$ be a collection of compact spin $\mathrm{I}$-holomorphic Lagrangian submanif…