33 problems

Let $M$ be a compact pre-quantizable symplectic manifold admitting a Hamiltonian action of a compact connected Lie group $G$. Guillemin–Sternberg's conjecture. Quantization commute…

Let $(M,\omega)$ be a positive monotone symplectic manifold of dimension six admitting a Hamiltonian circle action. A Fano manifold is a compact complex manifold whose anticanonica…

Let $K$ be a connected compact Lie group, let $M$ be a multiplicity-free compact Hamiltonian $K$-manifold with moment map $\mu:M\rightarrow\mathfrak k$, and fix a maximal torus…

Let $T$ act in a Hamiltonian fashion on a compact symplectic manifold $M$, with Duistermaat–Heckman measure on $\frak{t}^*$ obtained by pushing forward the Liouville measure under…

Let $(M,\omega)$ be a $2n$-dimensional closed symplectic manifold equipped with a Hamiltonian $S^1$-action whose fixed points are all isolated. Write $b_j$ for the $j$-th Betti num…

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a $2n$-dimensional compact symplectic manifold $M$, with moment map $\mu$ to the dual of the Lie algebra…

Isolated-fixed-point criterion. An $\mathbb S^1$-action is Hamiltonian if its fixed-point set is nonempty and isolated.

Let $(M,\omega,\phi)$ and $(M',\omega',\phi')$ be compact, connected complexity one $T$-spaces, with their moment maps, skeletons, and paintings. They have the same genus when the…

Let $G$ be a group, let $X$ be a $G$-space, and let $M=\mathrm{T}^*X$ be its cotangent Hamiltonian $G$-space, equipped with the appropriate anomaly trivialization data. Write…

A hyperspherical Hamiltonian action is a distinguished class of Hamiltonian boundary condition arising in the relative Langlands program. Relative Langlands conjecture. Hyperspheri…

Assume the setting of the Lie monoid action conjecture in Floer theory, with the actions Hamiltonian. Let $\mathcal{W}(T^*G)$ and $\mathcal{W}(T^*H)$ denote the wrapped Fukaya cate…

Let $\mathbf G$ be a group and let $\mathbf G^\vee$ denote its Langlands dual. Regard $(\mathbf G^\vee\curvearrowright\mathbf G^\vee\times\mathcal S^\vee)$ and…

Unimodality conjecture. If $(M,\omega)$ is a closed symplectically rational $8$-manifold admitting a Hamiltonian $S^1$-action, then the even Betti numbers of $M$ are unimodal.

Let $G$ be a complex reductive group, let $B e G$ be a Borel subgroup, and let $scrX$ be a hyperspherical $G$-variety with moment map bmucolonscrXtofg^. Define … where $\fb$ is t…

Let $X$ be a Liouville manifold equipped with a Hamiltonian $S^1$ action such that the fixed locus $X^{S^1}$ has codimension four and there are no finite non-trivial stabiliser gro…

Let $M$ be a compact symplectic manifold such that, for every $i$, its real cohomology in even degree satisfies … Suppose that $M$ admits a Hamiltonian $S^1$-action. Tolman's quest…

Non-commuting reduction by stages conjecture. The reduction by the product action should satisfy

Let $(M,\omega)$ be a six-dimensional closed monotone symplectic manifold equipped with an effective Hamiltonian circle action. A Kähler realization conjecture asserts that…

Let $M_{\cal T}$ be Tolman's manifold equipped with the symplectic form $\omega_{\lambda_1,\lambda_2}$, where $0<\lambda_1<\lambda_2$. A symplectic form is compatible with a Kähler…

Let $G$ act Hamiltonianly on a monotone symplectic manifold $X$. The action gives an $E_2$-algebra map … Lekili and Evans' construction gives a map from the endomorphisms of the co…

Kählerness conjecture. Then $(M,\omega)$ is $S^1$-equivariantly symplectomorphic to some Kähler manifold $(X,\omega_X,J)$ with some holomorphic Hamiltonian $S^1$-action.

Equivariant symplectic Fano conjecture. $M$ is diffeomorphic to a complex projective Fano $3$-fold.

Let $M$ be a compact connected symplectic manifold equipped with a symplectic circle action. Suppose that the action has isolated fixed points. McDuff's conjecture. The circle acti…

Let $(Z,\omega)$ be a six-dimensional symplectic Fano manifold with a Hamiltonian $S^1$-action. An integrable $S^1$-invariant complex structure compatible with $\omega$ is an integ…

Let $(M,omega)$ be a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points. Write $b_i(M)$ for the $i$th Bet…