130 problems

Let … where $z_i\in\mathbb{C}^*$ are pairwise distinct and $d_i\geq0$. Equivariant formality conjecture. The affine Springer fiber $\mathrm{Sp}_{\gamma}$ is equivariantly formal fo…

Let $G$ be a complex reductive group, let $B$ be a Borel subgroup, and let $W$ be the Weyl group. For $u,w\in W$ with $u\leq w$, write the equivariant Chern–Schwartz–MacPherson cla…

Let $\bar{\rho}=\bar{\rho}_{\theta,\theta'}$ be the sign-twist-independent map denoted by $\Phi$, from the negative half $\mathbb{Y}_\mathcal{Q}^-$ of the Yangian to the equivarian…

Let the affine Laumon space be the moduli space whose equivariant homology carries the proposed representation, and let an iterated $W$-algebra of type $A$ and an affine shifted Ya…

Ginzburg–Kapranov–Vasserot's conjecture. Every elliptic curve $A$ gives rise to a unique equivariant elliptic cohomology theory, natural in $A$.

Hyperkähler Kirwan surjectivity conjecture. The hyperkähler Kirwan map is surjective.

Braverman's conjecture. The partition function $Z(m)$ must be very closely related to the eigenfunction $Y(m)$. The conjecture proposes a relation between generating functions of i…

Let $K$ be a compact connected Lie group, let $c$ be a central value, and let $\mathcal{R}^{g}_{K}(c)$ denote the corresponding moduli or representation space for a genus-$g$ surfa…

Let $X=G/B$ be the flag variety of a complex semisimple group, with $B\supset T$ a Borel subgroup and maximal torus. Let $H_T^*(X)$ be the $T$-equivariant cohomology, viewed as a f…

Let $M$ be compact, and let $X$ be a Kähler manifold with a ${\mathbb C}^{\times}$-action, prequantum line bundle $L$, symplectic cuts $X_+$ and $X_-$, and reduced space $X_0$, wit…

Guillemin–Sternberg's conjecture. The equivariant holomorphic Euler characteristics should agree:

For each $t\in\mathbb{R}$, $X\in\mathfrak{g}$, let $\Theta(M,t)(X):=\int_M e^{-itd_X\lambda^{M_{\mathcal O}}}\alpha(X)$, and let $\Theta_0^{\mathcal O}$ denote the generalized-func…

Let $N$ be a possibly noncompact $G$-manifold, let ${\cal E}$ be a $G$-equivariant vector bundle on $N$ with $G$-invariant connection, and assume a $G$-invariant complex structure.…

Compatibility conjecture. The first two methods produce the same $Q$-algebroid and, although the third method produces a different $Q$-algebroid, it results in the same double comp…

Let $F/\mathbb Q$ be a $p$-adic Lie extension with Galois group $G$, let $\Lambda=\Lambda(G)$ be its Iwasawa algebra, let $T_p$ be a Galois-stable lattice, and put…

Let $G$ be a finite abelian group, let $V\to W$ be a $G$-cover as above, and let $X$ satisfy the hypotheses that make ${\rm H}^1(X,{\mathcal O}_X)$ $G$-isomorphic to…

Let $G$ be a finite abelian group and let $V\to W$ be a $G$-cover of smooth projective geometrically connected curves of positive genus over ${\bf Q}$. Set $A={\rm Jac}(V)$, let…

Canonical product conjecture. This canonical product always induces the correct product on $H^*(X)$, making the above identification an isomorphism of algebras.

Nakajima's conjecture. The equivariant Kirwan map is surjective for all quiver varieties.

Extension and fixed-point formula conjecture. (1) The class $\mu(C)$ extends to a class in

Pull-back conjecture. The square

Existence conjecture. There exists a complex oriented cohomology theory $E^*$ such that every closed oriented manifold is $E^*$-orientable and $E^*$ is $G$-sensitive for every fini…

Let $\boldsymbol\Lambda$ and $\mathbf S$ be the Koszul-dual algebra and coalgebra appearing above, and let weak $(\boldsymbol\Lambda,\mathbf S)$-modules and weak $\mathbf S$-comodu…

Let $G=(\mathbb Z/2)^r$ act freely on … and let $N=n_1+\dots+n_m$. Write $\pi_G:X/G\to BG$ for the classifying map of the action. Top-dimensional surjectivity conjecture. The induc…

Let $V$ be a symplectic vector space, let $G$ act symplectically on $V$, and let $X\to V/G$ be a symplectic resolution. The natural dilation action of $\mathbb C^*$ on $V$ lifts to…