14 problems

Let $M$ be a closed oriented manifold of dimension $n$, and let \H_{\mathrm{dR}} be its de Rham cohomology. Let \DBCyc\,\H_{\mathrm{dR}}[2-n] denote a suitable shifted cyclic c…

Let $M$ be a closed oriented manifold, and let $\StringH(\Loop M)$ carry the Chas–Sullivan string bracket $\StringOp_2$ and cobracket $\StringCoOp_2$. Let $\OPQ_{210}^\PMC$ and…

Let $\mathrm{C}^\infty(S^1)$ be the algebra of smooth complex-valued functions on the circle, and let $\operatorname{HE}^\ast$ denote entire cyclic cohomology. The de Rham homology…

Cyclic cohomology conjecture. The cyclic cohomology of $(H,R)$ satisfies

Cyclic cohomology conjecture. The cyclic cohomology of $A^{\infty}$ is given by

Let $A^{\infty}$ be the smooth subalgebra of the group $C^*$-algebra associated with the dihedral group, and let $HC^n(A^{\infty})$ denote its cyclic cohomology in degree $n$. Cycl…

Loday's proposal. These data should satisfy

Non-triviality conjecture. The cyclic cohomology class $[\mathfrak{c}_\omega]$ and the associated Hochschild cohomology class $[\mathfrak{h}_\omega]$ are non-trivial. These classes…

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

Duality conjecture. These dualities should also hold for the noncommutative smooth orbifold $\mathcal A_\theta \rtimes \Gamma$ under similar restrictions on $\theta$. This extends…

Let $L$ be an $n$-dimensional closed oriented Lagrangian submanifold of a symplectic manifold $(W,\omega)$, where $W$ is closed or convex at infinity, and let $H=H_{dR}(M)$ be its…

Let $M$ be a closed oriented manifold of dimension $n$, and let $H=H_{dR}(M)$ be its de Rham cohomology. The notation $B^{\text{cyc}*}H[2-n]$ denotes the shifted dual cyclic bar co…

Let ${\cal H}$ be a Hopf algebra, let $A$ be an ${\cal H}$-module algebra, let $C$ be an ${\cal H}$-module coalgebra acting on $A$ through a map … satisfying, for $h\in{\cal H}$,…

A characteristic map is a morphism from Hopf cyclic cohomology to cyclic cohomology induced by a modular pair in involution and an invariant trace. Characteristic-map component con…