31 problems

Landsman quantization reconstruction conjecture. The classical limit functor is almost inverse to the quantization functor defined by Landsman, in the sense that the classical limi…

Quantizability conjecture. Any Lie bialgebroid is quantizable.

Let $N$ be a smooth manifold such that its tangent bundle admits a CYMH GT, and let $K\to S$ be a Lie algebroid bundle over a smooth manifold $S$ which also admits a CYMH GT. Set ……

Lie's second theorem. For any morphism of Lie algebroids $\phi:A\to B$, there is a unique morphism $\Phi$ from the Weinstein groupoid $\mathcal{H}(A)$ to any Weinstein groupoid…

Uniqueness conjecture. $\mathcal{H}(A)$ is the unique source-simply connected Weinstein groupoid whose Lie algebroid is $A$.

Let $A$ be an integrable Lie algebroid, let $A^*$ be its dual equipped with its natural Lie–Poisson structure, and let $G$ be a Lie groupoid integrating $A$. A deformation quantisa…

Let $X_0$ be the base of a pre-realization of a cofoliation, let $E$ be the corresponding vector bundle with anchor $a_{X_0}:E\to T_{X_0}$, and set $N=\ker a_{X_0}$. Bracket-differ…

Weinstein's conjecture. This topological groupoid has some differentiable structure.

Let $A$ be a Lie algebroid that admits a proper integrating groupoid $\mathcal{G}$ whose $s$-fibers are $2$-simply connected. Cohomological rigidity conjecture. Then … This conject…

A Lie algebroid on a manifold is given, and one considers a proposed construction that produces a Lie algebroid on its loop space. A Dirac structure is the relevant Lie algebroid i…

Landsman's conjecture. The quantization of $A^*$ is related to the groupoid $C^*$-algebra of $\mathcal G$.

Let $\mathcal{A}\to\mathsf{WfMd}$ be a Lie algebroid of controlled intervention modes over a typed Markdown workflow space, with anchor $\rho:\mathcal{A}\to T\mathsf{WfMd}$. For co…

Lurie–Pridham extension conjecture. The Lurie–Pridham theorem should extend to a theorem relating formal deformation of derived schemes to Lie algebroids over them.

Toën and Vezzosi's conjecture. The assignment from such dg-Lie algebroids to derived foliations is fully faithful.

Higher smooth homotopy hypothesis. Lie $n$-algebroids up to $n$-equivalence are equivalent to source $n$-connected Lie $n$-groupoids up to equivalence. In particular, generalized…

Smooth homotopy hypothesis. After localizing with respect to these $n$-equivalences, there is an equivalence between two categories, one groupoid-like and one algebroid-like.

Cohomological invariance conjecture. One has

Invariance conjecture. There is an equivalence of categories between representations up to homotopy of length at most $n$ of $A$ and of $B$.

Integrability converse. The Lie algebroid $\mathfrak{g}\to X$ is integrable if and only if it is equivalent, as an LA-groupoid, to a simple foliation.

Modular class conjecture. The modular class measures the obstruction to the existence of a volume form on the differentiable stack associated to the groupoid integrating the Lie al…

Let ${\mathfrak{g}}$ be a Lie algebra, and let $\int {\mathfrak{g}}_\bullet$ be the simplicial Banach manifold whose $k$-simplices are the differential graded algebra morphisms … E…

In general relativity, the Noether charges are conserved quantities associated with spacetime diffeomorphism symmetry. A hamiltonian Lie algebroid is a Lie algebroid equipped with…

Bracket-compatibility conjecture. The section $\mu$ is a bracket-compatible momentum section with respect to the trivial connection.

Androu–Zambon conjecture. Not every smooth singular foliation is of this type, not even in a neighborhood of a given point.

Let $A\to M$ be a transitive Lie algebroid, and let $p:A\to A^\mathrm{ab}$ be the morphism to its abelianization, characterized by the universal property of abelianization. Surject…