47 problems

Let $r,N>0$, let $S=\mathbb{C}^2$, and set $E=\bigoplus_{i=1}^N\mathcal{O}_S\langle y_i\rangle$ and $V=\bigoplus_{i=1}^r\mathcal{O}_S\langle v_i\rangle$. Strong Segre symmetry. The…

Laurent-polynomial conjecture. For every degree $d$, $\Omega_{d[L]}$ is a Laurent polynomial in $q_0$, $q_1$, $q_2$, and $q_3$.

Let $G$ be a finite group acting effectively on a smooth quasiprojective variety $X$. For $g\in G$, let $X^g$ denote the invariant subvariety and let $C(g)$ be the centralizer of…

Nonnegative-coefficient conjecture. For every $m\geq 1$, $\mathcal{N}^{(m)}_n(q,t)$ is a polynomial in $q$ and $t$ with nonnegative integer coefficients. The authors further conjec…

Higher-degree denominator conjecture. For every integer $d>1$,

Hebey–Vaugon conjecture. If $(M^n,g)$ is not conformal to the standard metric on $S^n$, or if the action of $G$ has no fixed point, then

Equivariant torus-fibration mirror-symmetry conjecture. There should be equivalences

Let $R_{\sigma\tau}^{(n)}(t,\lambda)$ be the coefficient of $\hbar^n$ in the asymptotic solution matrix $R_{\sigma\tau}(t,\lambda,\hbar)$ of the equivariant quantum differential eq…

Let $G$ be a finite group, or more generally a linear algebraic group, acting on a variety $X$, and let $X\to Y:=X/G$ be the quotient. A $G$-variety is a variety with a $G$-action,…

Local structure conjecture. (1) There exists a proper continuous map

Let $X$ be a quasiprojective scheme over a field $k$, let $G$ be a finite group acting on $X$, and assume that $\operatorname{char} k$ does not divide $|G|$. Let…

Assume Setup, with $X$ a Fano manifold and $\omega \in c_1(X)$. Let $v \colon P \to \mathbb{R}_{>0}$ be a weight, let $\mathscr{V}$ be a canonical extension sheaf with extension cl…

Let Setup be fixed, let $v \colon P \to \mathbb{R}_{>0}$ be a weight, and let $\mathcal{E}$ be a reflexive sheaf of rank $r$ that is $v$-stable. Bogomolov–Gieseker conjecture. The…

Let Setup be fixed. For a weight $v \colon P \to \mathbb{R}_{>0}$, let $C_v>0$ be a constant, and let $\mathcal{E}$ be a $\mathbb{G}$-equivariant torsion-free sheaf. The equivarian…

Let $G_{\mathrm{data}} \subseteq G$ act on $M$, and let $\Gamma$ be a continuous boundary whose symmetry group contains $G_{\mathrm{data}}$. For a target precision $\varepsilon>0$,…

Let $M^{n+1}$ be as in the equivariant setting, with a compact Lie group $G$ acting by isometries and satisfying … A $G$-homology class is represented by a fixed $G$-hypersurface…

Let $M^{n+1}$ be a closed Riemannian manifold equipped with a compact Lie group $G$ acting by isometries, satisfying … A closed embedded minimal hypersurface is $G$-invariant when…

Let $G$ be an algebraic group, let $X$ be a smooth projective variety with a $G$-action, and let $\pi:X\to Y$ be a $G$-contraction. Let $\mathrm{Stab}(X_G)$ denote the space of sta…

Let $\theta:\mathcal{M}\to\mathbb{A}$ be an integrable system, with virtual and genuine equivariant multiplicities $\mu_F(t)$ and $m_F(t)$ for fixed components $F$. Wobbly-componen…

Equivariant Yau conjecture. The manifold $M$ contains infinitely many closed embedded minimal $G$-hypersurfaces $\Sigma$ in the given $G$-homology class. This is the equivariant ge…

Let $G$ act on a surface $X$. An atom is a component in the atomic semi-orthogonal decomposition associated with the $G$-surface, and an atom is non-twisted permutation if it has t…

Relative integral identity. The equality

Let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime. Let $\mathcal{H}_{m,n}$ and $\mathcal{H}'_{m,n}$ be the derived objects obtained from the ordinary and eccentric flag commuting…

Let $X$ be a Calabi–Yau 4-fold with an algebraic torus $T$ acting while preserving the Calabi–Yau volume form, and let $L$ be a $T$-equivariant line bundle on $X$. For…

Quantum K Whitney relations. For $j=1,\dots,k$, the following relations hold in $\operatorname{QK}_T(X)$: