24 problems

Let $\mathcal{C}$ be a smooth and compact $A_\infty$-category. Its noncommutative Hodge–de Rham spectral sequence is the spectral sequence associated with the cyclic-homology filtr…

An $A_{\infty}$-category of lagrangian submanifolds in a symplectic manifold is the category whose objects are such lagrangian submanifolds and whose morphisms and compositions hav…

Let ${\cal C}$ be an $A_{\infty}$-category such that $HH^1({\cal C})$ is abelian and the connected automorphism group $G_{\cal C}=Aut^0({\cal C})$ is a finite-dimensional commutati…

Let a directed $A_{\infty}$-pre-category be an $A_{\infty}$-pre-category whose objects are indexed by ${{\bf Z}}$ with transversal sequences ordered by the indices, and consider…

Let an $A_{\infty}$-pre-category be understood with the equivalence relation defined in the source, and let an $A_{\infty}$-category with strict identity morphisms be a non-unital…

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…

Let $a_n,\ldots,a_1$ be a sequence of morphisms represented by a polygon with exactly one boundary whose winding number is $-2$. Fix a morphism $a_n$ in each such sequence, and wri…

Let $\Lambda$ be a Legendrian link with marked points, let a Morse function be chosen, and let $\mathcal{A}ug_+$ and $\mathcal{C}$ be the associated strictly unital proper…

A-infinity enhancement conjecture. The category $\mathcal{F}(W,G)$ becomes an $A_\infty$-category, and the above quasi-isomorphism extends to an $A_\infty$-equivalence of categorie…

Relative Fukaya category conjecture. The full subcategory of $\mathcal{W}(M,D)$ whose objects are compact Lagrangians is filtered quasi-equivalent to $\mathcal{F}(M,D)$.

Vanishing-obstruction conjecture. Such obstructions always vanish when $\rho$ comes from the monodromy of an isolated singularity.

Let $\Lambda^{{\mathbb Q}}_+$ be the positive Novikov subring, let $\frak L=\{L_i\mid i=1,\dots,N\}$ be a finite set of relatively spin compact Lagrangian submanifolds of…

For a tuple $a=(a_1,\ldots,a_n)$, let $-a=(a_2,\ldots,a_n)$, and let $\mathcal{A}_a$ be the directed Fukaya–Seidel $A_\infty$-category associated with the chain polynomial indexed…

Flag Hilbert scheme deformation conjecture. There is an $A_\infty$-deformation $(\mathcal{A}_\infty(FHilb_n),\tilde{\otimes})$ of $(\mathrm{DG}(FHilb_n),\otimes)$ such that

Let $(M_1,\mathcal E_1)$ and $(M_2,\mathcal E_2)$ be composable $3$-dimensional cobordisms as in part (A-3) of the preceding conjecture, and let $(M,\mathcal E)$ be their compositi…

Let $(M,\mathcal E)$ be a cobordism $(\Sigma_1,\mathcal F_1)\to(\Sigma_2,\mathcal F_2)$, where either boundary component may be empty, and let $\mathfrak I_{(M,\mathcal E)}$ be the…

Fix a commutative ring $R$, a group $G$, and a $G$-bundle $\mathcal F$ over a Riemann surface $\Sigma$. Let $\mathfrak I(\Sigma,\mathcal F)$ denote the proposed unital filtered…

Let $\mathscr C$ be a strict filtered $A_{\infty}$ category, and let $(\frak c_i)$ be a Cauchy sequence of objects of $\mathscr C$ with respect to the Hofer distance. Suppose the i…

Let $(X_i,c9_i)$ be compact spin symplectic manifolds, and let $L_1$ and $L_{12}$ be spin immersed Lagrangian submanifolds of $(X_1,c9_1)$ and $(X_1X_2,-_1^*(c9_1)+_2^*(c9_2))$, re…

Let $\EuC$ be a proper and smooth $A_\infty$ category. Its negative cyclic complex $CC^-_\bullet(\EuC)$ carries the Hodge filtration, which induces a spectral sequence. We say that…

Kontsevich–Costello's conjecture. The Fukaya category of a Calabi–Yau manifold should be a Calabi–Yau $A_\infty$ category: it should carry a non-degenerate symmetric bilinear pairi…

Let $(M,\omega)$ be a symplectic manifold, let $V\subset M$ be a Lagrangian submanifold diffeomorphic to $\mathbb{C}\mathbb{P}^n$, and let $L$ be another Lagrangian object of the F…

Kontsevich–Soibelman conjecture. Quasi-equivalence classes of $A_{\infty}$-pre-categories over $k$ are in bijection with quasi-equivalence classes of $A_{\infty}$-categories over…

Twisted-complex model conjecture. There is an equivalence of triangulated categories