12 problems

Let $M$ be a symplectic manifold that admits a toric structure. An almost toric fibration of $M$ may contain nodal fibers obtained by nodal trades, while a toric fibration has only…

Let two toric manifolds, equivalently smooth toric varieties, be given. Their integral cohomology rings are isomorphic if and only if the manifolds are diffeomorphic. Masuda–Suh co…

Let $\mathcal{B}$ be a venustus building set, and let $X^\mathbb{R}_{\mathcal{B}}$ be its associated real toric manifold. Write $\beta_k(X^\mathbb{R}_{\mathcal{B}})$ for its ration…

Donaldson's conjecture. The Abreu equation

Let $X(A)$ and $X(A')$ be Bott manifolds of complex dimension $n$. Their integral cohomology rings are graded rings of the form $H^*(X(A))$ and $H^*(X(A'))$. Bott manifold cohomolo…

Let $M$ be a toric simply connected 4-dimensional manifold, and consider an $\mathcal{N}=2$ supersymmetric gauge theory on $M$ with partition function defined by localization. Tori…

Let $V$ be a compact toric contact manifold. A contact manifold is orderable if its identity component of the contactomorphism group admits no positive contractible loop, and a pre…

Strong cohomological rigidity conjecture for Bott manifolds. Any graded ring isomorphism between the cohomology rings (with integer coefficients) of two Bott manifolds is induced f…

Seidel-element and Jacobian-generator conjecture. The isomorphism

Let $X$ be a compact toric manifold and let $L(u)=\pi^{-1}(u)$ for $u\in\operatorname{Int}P$. A toric fiber is non-displaceable if no Hamiltonian diffeomorphism $\psi:X\to X$ satis…

Let $X$ be a compact toric manifold, let $\textbf{i}$ be an idempotent of quantum cohomology corresponding to an indecomposable factor, and let $u(1,\textbf{i})$ denote the fiber d…

Let $X$ be a compact toric manifold, let $\textbf{i}$ be an idempotent of quantum cohomology corresponding to an indecomposable factor, and let $L=L(u(1,\textbf{i}))$ be a Lagrangi…