23 problems

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.

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

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…

The motivic $\mathbf{A}^1$-homotopy contraction conjecture. The inclusion $Y\to U$ is an $\mathbf{A}^1$-homotopy equivalence, that is, an isomorphism in the $\mathbf{A}^1$-homotopy…

Let $d\geq 2$ be an integer. Let $A\subset\mathbb{T}^d$ be an orbit closure under the $\times 2,\times 3$ actions. Orbit-closure conjecture. Then $A$ is a union of finitely many (p…

A flag Gorenstein$^*$ $n$-complex $K$ is assumed to satisfy the SCC, and the NSC is the condition that there is no full subcomplex $K_I\subset K$ such that $K_I$ is a suspension an…

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…

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…

Maximal symmetry rank conjecture. One has

Let $F_n=\mathbb{K}\langle x_1,\ldots,x_n\rangle$ be the free associative algebra, and let $\mathbb{T}_{n-1}$ act effectively on $F_n$. Free associative Białynicki-Birula conjectur…

Let $k$ be an algebraically closed field of characteristic $2$, let $S=k[x_1,\ldots,x_r]$ be the polynomial algebra in $r$ variables of degree $-1$, and let $(M,\partial)$ be a dif…

Rigidity conjecture. $M^n$ is equivariantly diffeomorphic to an effective, linear action of $T^k$ on a manifold of one of the following forms:

Guillemin–Ginzburg–Karshon's conjecture. An element $\beta\in\Omega_*^{U,G}$ is zero if and only if all integral equivariant cohomology Chern numbers of $\beta$ vanish.

Generalized affine-paving conjecture. If $\mathfrak{o}$ is a total order such that $P^{\mathfrak{o}}(t)=P_V(t)$, then the decomposition of $V$ according to $\mathfrak{o}$ is a gene…

Formal Poincare polynomial conjecture.

Let $M^{2m-1}$ be a closed Riemannian manifold with positive sectional curvature. Let $q$ denote the smallest prime divisor of $m$, and let an $r$-torus act effectively and isometr…

Equivariant cohomological realization conjecture. For every semilocal $\mathcal T$, there is an isomorphism of graded $S[\mathfrak g^*]$-modules

Let $X$ be a nilpotent finite CW-complex. Its rational torus rank ${\operatorname{rk}}_0(X)$ is the largest integer $n$ for which a torus $T^n$ acts almost freely on a finite CW-co…

Toral rank conjecture. If $T^r$ acts almost freely on $X$, then

Let $\Gamma$ be as in the uniformly partially hyperbolic quasi-affine conjecture, and let $\mathbb T^n$ be the $n$-torus. A toral action is weakly hyperbolic when it has the weak h…