35 problems

Unitality–homotopy unitality equivalence. An $A_\infty$-morphism is unital if and only if it admits a homotopy unital structure; equivalently, any unital $A_\infty$-morphism admits…

An extended deformation problem of a holomorphic vector bundle is described by the Maurer–Cartan equation on spaces such as $\Omega^{0,*}(X,\operatorname{End} E)$,…

$A_\infty$-Formality conjecture. The associative differential graded algebra $D^\bullet_\mathrm{poly}(\mathbb R^d)$ is quasi-isomorphic, as an associative algebra, to its cohomolog…

Let $A$ be a homologically smooth, compact, finite-dimensional $A_{\infty}$-algebra. A non-degenerate cohomology class $[\varphi]$ of degree $\dim A$ is a class defining the corres…

Crystalline comparison conjecture. These two complexes are quasi-isomorphic for all $n,m\geq 1$ and all primes $p$; if $A$ is also compact, the homology of either complex is flat o…

Let $\mathcal{H}$ be the bialgebra object, let $\mathcal{H}_{\infty}$ be the canonical differential graded submodule resolving its prematrad version…

Deformation-theoretic conjecture. There exists a deformation theory for $A_{\infty}$-bialgebras in which the infinitesimal deformations of classical differential graded bialgebras…

Let $E_0$ and $E_1$ be the objects appearing in the surgery-formula construction, and let $\operatorname{End}_{w\mathrm{Fil}}(\beta_0^{E_0}\oplus\beta_1^{E_1})$ denote the endomorp…

Let ${\mathrm{K}}_m^n$ be the extended Khovanov arc algebra indexed by $m$ and $n$, and let $\operatorname{HH}^2_{i-2}({\mathrm{K}}_m^n,${mathrm{K}}m^n)…

Suppose $(X,\omega)$ and $\mathcal D$ are as in the beginning of the introduction, and let $L\subset X\setminus\mathcal D$ be a relatively spin compact Lagrangian submanifold. Let…

De Kleijn–Wierstra's conjecture. The nerve and the simplicial Maurer–Cartan set of the commutator algebra should be homotopy equivalent:

Projective resolutions of standard modules are interpreted using certain Heegaard diagrams. Holomorphic-disk conjecture. Holomorphic disk counts in these diagrams should encode bot…

Let $W$ be a Coxeter group with realization $V$ over $\mathbb{C}$, and let $\mathcal{C}(W)$ denote the corresponding Soergel Hecke category. Let $\operatorname{Tr}$ be its universa…

Let $A$ be a differential graded algebra (dga), let $\omega_A\in A$ be an even-degree element, and define … Assume that $A$ has an $A_\infty$-minimal model with higher operations…

Let $M$ be an orientable $N$-dimensional $k$-connected compact manifold, with $\dim H^{k+1}(M)=1$. Let $l\geq 3$ and assume … An $A_\infty$-minimal model of $\Omega^*(M)$ is a mini…

Fukaya's conjecture. Witten's isomorphism can be enhanced to an isomorphism of $A_\infty$ algebras, or of $A_\infty$ categories.

Let $(X,[?]$

Partially wrapped Floer–Legendrian graph correspondence conjecture. There is an $A_\infty$ quasi-isomorphism between the partially wrapped $A_\infty$ algebra…

Let $\Pi_n$ be the $n$-dimensional pair of pants, let $L_I$ be the relevant objects, and consider the algebra $\bigoplus_{I,J}HW^*(L_I,L_J)$. Assume its $A_\infty$-structure is com…

Infinity-contraction algebra reconstruction conjecture. The completions of the stalks at $p$ and $q$ are isomorphic if and only if $A^\infty_{con}$ and $B^\infty_{con}$ are derived…

Figure-eight bubble conjecture. The moduli space of figure-eight bubbles should give a bounding cochain, in the sense of Fukaya–Oh–Ohta–Ono, for the filtered $A_{\infty}$ algebra a…

Open quantum Kirwan map conjecture. Counting affine vortices over the upper half-plane $\mathbb H$ defines an $A_\infty$ morphism

Let $\mathbb{K}$ be a ground field of characteristic zero, let $\Gamma$ be a tree of type $E_6$, $E_7$, or $E_8$, and let $A_\Gamma$ be the corresponding zigzag algebra. Exceptiona…

Knizhnik–Zamolodchikov associator conjecture. The universal $A_{\infty}$ structure constructed in the paper corresponds to choosing the Knizhnik–Zamolodchikov associator.

Let $S$ be the underlying scheme, and let $\mathcal{B}$ be a sheaf of $\mathbb{Z}$-graded differential-graded $\mathcal{O}_S$-algebras and $\mathcal{A}$ a sheaf of $\mathbb{Z}$-gra…