71 problems

Equivariant positivity conjecture. The coefficient $c_{v;w}(\alpha)$ is a polynomial in the positive roots $\alpha$ with non-negative coefficients. Numerical evidence supports this…

Let $R_n$ be the ring defined in the surrounding construction, with binary operation $\star$. Associativity conjecture. $(R_n,\star)$ is an associative ring. The supplied context d…

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 $F/\mathbb Q$ 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 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…

Pull-back conjecture. The square

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…

Let $\Gamma$ be a graph with axial function $\alpha$. For each vertex $p$, let $\sigma_p$ be its index, let $F_p$ be the set of vertices that can be joined to $p$ by an ascending p…

Let $\Gamma$ be a graph with axial function $\alpha$, let $H(\Gamma,\alpha)$ be its equivariant cohomology ring, and let ${\mathbb S}({\mathfrak g}^*)$ be the polynomial ring occur…

Let $\lambda$ and $\mu$ be partitions, let $\mathbf x_k=(x_1,\ldots,x_k)$, and let $s_\lambda(\mathbf x_k;\mathbf y)$ denote the double Schur polynomial. Write $\lambda\vee\mu$ and…

Let $u,v\in S_n$ be vexillary permutations, let $\phi(w)$ be the Wachs flag of $w$, and write $a\preceq b$ when $a$ is a prefix of $b$. Let $\mathfrak S_w(\mathbf x;\mathbf y)$ be…

Let $X$ be a type A Nakajima quiver variety with its natural torus action, let $p$ be a $\mathsf{T}$-fixed point, and let $\boldsymbol{a}\to 0$ denote the non-equivariant limit. Qu…

Let $X=T^*Gr(k,n)$, let $\boldsymbol{a}=(a_1,\ldots,a_n)$ be the equivariant parameters, and let $\boldsymbol{a}\to 0$ denote the non-equivariant limit. For a partition $\lambda$ in…

Homological Hikita conjecture. There is an isomorphism of graded $\mathbb C[(\mathfrak t_\bv/S_\bv)\times\ff]\otimes\mathbb C[\hbar]$-algebras

Let $Q$ be the quiver, let $d$ be the dimension vector, and let $\delta$ be as in Theorem; write $\operatorname{supp}\delta$ for its support, let $d_i$ denote the components of $d$…

Let $X$ and $X^!$ be symplectically dual, and let $\mathcal{A}$ be the universal quantization of $X$. The quantization exact sequence and the equivariant cohomology exact sequence…

Let $X$ and $X^!$ be symplectically dual, and let $Y$ be the affinization of $X$. Let $Y^T$ denote the scheme-theoretic fixed locus of $Y$ under the $T$-action, with grading induce…

Intrinsic-limit conjecture. Laurent-deformed Calabi–Yau hypersurfaces closed or completed by the intrinsic limit are not algebraic varieties; they are toric spaces equipped with a…

Let $G$ be a compact Lie group acting on a manifold $X$, let $X{/{\!/}^{\nabla}}G$ denote the quotient stack encoding the action with connection data, and let…

Nonnegativity conjecture. For all $u,v,w$, $c_{uv}^w(y;z)$ is a polynomial in the differences $y_i-z_j$ with nonnegative integer coefficients.

Let ${\mathscr{F}}$ be a geometric oriented abelian sheaf over a derived stack $\mathsf{S}$. Write $\operatorname{Sp}_{\mathrm{fin}-\mathrm{gl}}$ for the symmetric monoidal…

Let $R$ be the relevant elliptic root system, let $\delta_E$ and $\delta_{\mathrm{pt}}$ be the indicated imaginary-root classes, and let…

Let $L_f$ be the link of the plane curve germ $f=0$. Let $\widetilde{\operatorname{Quot}}^{\vec{x},\vec{y}}$ be the bigraded $\mathbb{C}[\vec{x},\vec{y}]$-module assembled from the…

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…