25 problems

Let $A$ be an associative algebra with product $m$, and let $varphi$ be a $k$-cocycle. The infinitesimal deformation $m_t=m+tvarphi$ determines a cycle in the homology of the graph…

Data-loss conjecture. In passing to the algebraic closure, one kills off additional data within the homotopy type, namely the data that depends on larger primes.

Let $d=d_2+d_k$ be a codifferential, and let $d_e=d_2+d_k+d_l+\cdots$ and $d_e'=d_2+d_k+d_l'+\cdots$ be extensions. Two codifferentials are formally equivalent when a formal automo…

Let $X$ be a homotopy differential graded non-unital algebra, and let $X(1)$ be the underlying chain complex of the associated homotopy semigroup. The construction described in the…

The lifted complex Morse model conjecture. The induced deformation $R_{\mathfrak C_S,\gamma_S}$ is $A_\infty$-quasi-isomorphic to the directed total algebra $R_S$:

Let $M$ be a smooth manifold, let $n$ be a positive integer, let $[g,\mathcal{A}]$ be an $(n{-}1)$-gerbe with full connection, and let $H\in\Omega^{n+1}_{\rm cl}(M)$ be the field s…

In the BV formalism, let $I$ be a local interaction for a UV-finite theory, and let $r\to 0$ denote the removal of the regularization. Write $l_n^\hbar$ for the multilinear operati…

Let $(F,Q,(-,-),W)$ be a free BV field theory, and suppose its Green's witness $W$ is obtained by truncating an $L_{\infty}$ algebra to its underlying cochain complex. Green-witnes…

Locality conjecture. The full package of chain-level string topology, in particular the $\infty$-involutive Lie bialgebra (or $IBL_{\infty}$-algebra) structure on the $S^1$-equivar…

Algebraic interpretation conjecture. The first coefficients of the beta function can be interpreted as algebraic operations in some $A_\infty$-algebra.

$L_{\infty}$-algebra conjecture. There exists an $L_{\infty}$ structure on the complex $({\bf F}^{\bullet}_{{\bf b}^-},\mathcal{Q})$, such that its bilinear operation is the symmet…

A Poisson manifold carries a Poisson gauge theory, with gauge symmetries acting on its fundamental fields, and an $L_\infty$ algebra is used to encode the gauge structure. L-infini…

Functoriality conjecture. There is an $L_\infty$ morphism from the $L_\infty$ algebra $((\mathcal{A}^+)^{\operatorname{comm}},\{\ell_k^+\})$ of $\Lambda^+$ to the $L_\infty$ algebr…

Invariance conjecture. The homotopy Poisson algebra $(\mathcal{A}^{\operatorname{comm}},\{\ell_k\})$ of a pointed Legendrian link $\Lambda$ is invariant under Legendrian isotopy of…

Assume $\Bbbk$ has characteristic zero and Hypothesis … can be taken to be $\widetilde{SC}_{\Lambda}=SC_{\Lambda}$ with differential $\partial=\ell^1_\beta$. This would eliminate t…

Let $F\colon\mathcal A\to\mathcal B$ be a homological epimorphism of $A_\infty$-algebras, and let $\widetilde{\mathcal A}$ be an $E_n$-algebra enhancing $A$. The $E_n$-enhancement…

The paper constructs invariant $n$-multiplications on quaternionic algebras, including the quaternionic algebra $\textit{{\fontencoding{OT2}\fontfamily{wncyr}\selectfont Zh}}$ and…

Let $A$ be an $A_\infty$ algebra over a characteristic-zero field with a cocommutative, conilpotent coproduct $\Delta$ that is a strict $A_\infty$ morphism … Let $\overline\Delta$…

Let $L$ be an $L_\infty$ algebra. Write $\mathcal C(L)$ for its Chevalley–Eilenberg coalgebra, $\mathcal U L$ for Lada and Markl's universal enveloping $A_\infty$ algebra, and…

String-topology chain-model conjecture. This construction gives a chain model for Chas–Sullivan string topology.

Let $M$ be a supermanifold with a homotopy Poisson structure. This structure induces higher Koszul brackets on the space of forms $\Omega(M)$ and a Schouten bracket on the multivec…

Joyce–Song formula. The following identities should hold:

Canonical $L_{\infty}$ morphism conjecture. There is a canonical $L_{\infty}$ morphism from $\Omega^{\bullet}_{\mathbb J,H}$ to the total complex of, extending the given quasi-isom…

Let $M$ be a manifold, let $\mathfrak{g}$ be the $L_{\infty}$-algebra associated to a Sullivan model of $M$, let $\mathbb{S}(\mathfrak{g})$ denote its strictification, and let…

Operadic higher-order derivation conjecture. The related $\mathcal P^!_\infty$-braces should lead to a reasonable notion of higher-order derivations of ${\mathcal P}$-algebras.