55 problems

Let $X$ be a finite-dimensional topological space, and let $T^k=(S^1)^k$ be a $k$-dimensional torus acting almost freely on $X$. The total Betti number is…

Let $M$ be a manifold and let $X$ be a globally hypoelliptic vector field on $M$. A vector field on a torus is called constant diophantine when it is given by a constant vector sat…

Let $M$ be a closed symplectic manifold admitting a Hamiltonian torus action with isolated fixed points. The Hard Lefschetz conjecture. The manifold $M$ satisfies the Hard Lefschet…

For $n\geq 1$, let $P_n=\mathbb{K}[z_1,\ldots,z_{2n}]$ be the commutative Poisson algebra with bracket … An action is effective if its kernel is trivial, and regular if it is algeb…

Let $F_n=\mathbb{K}\langle x_1,\ldots,x_n\rangle$ be the free associative algebra, and let $\mathbb{T}_{n-1}$ be the $(n-1)$-dimensional algebraic torus. Free-associative torus-act…

Let $X$ be a smooth compact complex algebraic variety with a semifree action of the algebraic $1$-torus $\mathbb{C}^*$, and suppose that the action has a positive number of isolate…

Białynicki-Birula–Sommese conjecture. Part of the algebraic theory of GIT quotients should extend to the more general case of meromorphic actions of $({\mathbb C}^*)^k$ on a reduce…

Consider an algebraically integrable system with a singular orbit $O$. An orbit is of finite type if, in a small complexification, the connected component of the common level set c…

Let $T$ be a torus and let $\leq i\leq \dim T$. For each subtorus $\tau\subset T$, let $\mathcal{A}_\tau$ denote the corresponding summand of the module $\mathcal{A}$, and set … Le…

Non-contractibility conjecture. Then for every non-Hamiltonian subcircle action of $T$, the orbits of the circle action and all of their iterates are non-contractible.

Let $G$ be a complex reductive group with Weyl group $W$, and let $X_{\mathfrak{g}}$ be one of the varieties attached to its reductive Lie algebra $\mathfrak{g}$. Consider the toru…

Let $M^{2n}$ be a GKM$_4$ manifold with GKM graph $\Gamma$. A torus manifold is a manifold with an effective action of a torus whose dimension is half the manifold dimension, and a…

Let $M\in\mathbb{Z}^{d\times n}$ satisfy $0\in\operatorname{int}(P)$, let $v\in(\mathbb{Q}(\operatorname{i})^\times)^n$, and let $\varepsilon\in\mathbb{Q}_{>0}$. Let…

Polynomial-time minimizer approximation conjecture. A point $x\in\mathbb{R}^d$ that is $\varepsilon$-close to $\arg\min f(x)$ in Euclidean distance can be computed in polynomial ti…

Let $T={\mathbb G}_m^r$ be a torus over an algebraically closed field acting on a smooth proper variety $X$. Assume that there is a $T$-linearized line bundle $\mathcal{L}$ on $X$…

The DT–PT virtual structure sheaf localization conjecture. There exists a canonical halving $\mathbb{E}_{1/2}$, given explicitly in the source's Definition, such that

Let $\lambda_1,\ldots,\lambda_s$ be partitions with total size $k$, and let $\mathrm{Hilb}^{\lambda_1,\ldots,\lambda_s}(\mathbb{C}^n)\subset\mathrm{Hilb}^{[k]}(\mathbb{C}^n)$ be th…

Let $\mathrm{Hilb}^k(\mathbb{C}^n)$ be the Hilbert scheme of length-$k$ subschemes of affine $n$-space, with its natural torus action, and let $\mathrm{CHilb}^k_0(\mathbb{C}^n)$ de…

Let $X$ be a nilpotent finite-dimensional CW-complex admitting an almost free action by the torus $T^n$. Write $H(X;\mathbb{Q})$ for its rational cohomology. Torus rank conjecture.…

Nonextendibility conjecture. Most homogeneous GKM-manifolds are nonextendible, with possibly a few exceptions determined by the condition

Gap-property conjecture. Any homogeneous GKM-manifold satisfies the gap property: if its GKM-graph is $j$-independent for some $j\geq 4$, then it is $n$-independent, where $n$ is t…

Spin torus-action conjecture. The manifold $M$ is homeomorphic to one of $S^3$, $S^4$, $S^5$, or

Let $I_{\lambda}$ be a non-Borel monomial ideal of $\Bbbk[X_1,X_2,X_3]$ with extra dimension $6$. Let $\widehat{G}(2,6)$ be the affine cone over the Grassmannian $G(2,6)$, and call…

Let $(\Lambda V,{\operatorname{d}})$ be a minimal Sullivan algebra. Let … be a relative minimal model with $\deg t_i=2$ for $1\leq i\leq r$, and suppose that … is finite dimensiona…

Boundedness conjecture. The point $x\in X$ always specializes to a bounded family of singularities with a torus action.