60 problems

Let $G$ be a simple Lie group with Borel subgroup $B$, let $G/B$ be its flag variety, let $P$ be the weight lattice, and let $R$ be the coefficient ring used in the quantum deforma…

Quantum K-theory Chevalley conjecture. For every $w\in W$,

Finite-difference Toda eigenfunction conjecture. $I_K$ is a common eigen-function of the conservation laws of the finite-difference Toda lattice corresponding to ${\mathfrak g}'$.

Let $X$ and $X^!$ be symplectically dual varieties arising from Coulomb-branch and resolved Higgs-branch constructions for the gauge theory $(G,N)$. Let $\zeta$ be the relevant roo…

Quantum K-theoretic Hikita conjecture. This $(\hbar=q)$-specialization should be equal to the $D$-module of graded traces for the quantized K-theoretic Coulomb branch.

Let $F\ell_n$ be the complete flag variety, let $S_n$ be its Weyl group with simple reflections $s_1,\ldots,s_{n-1}$, and let $QK(F\ell_n)$ denote its quantum $K$-theory with quant…

Gerbe decomposition conjecture. For the Ruan–Zhang/Chern–Simons level corresponding to ordinary quantum $K$-theory,

Flag-manifold relations conjecture. For $\ell$ in the geometric window, the $\prod_i S_{k_i}$-invariant combinations of the Coulomb-branch equations generate the ideal of relations…

Grassmannian relations conjecture. Symmetric combinations of the equations

Symmetrized Coulomb-branch relations conjecture. The symmetrizations of the Coulomb-branch equations are relations in the appropriately twisted quantum $K$-rings.

Mirror-triviality conjecture. A level $\ell$ is in the physical geometric window if and only if

Level-structure dictionary. The same result should hold for quantum $K$-rings with level structures, with the levels related by

Let $G$ be a connected, simply-connected, simple linear algebraic group over $\a0\mathbb{C}$ of Lie type $A$, let $P$ be a parabolic subgroup containing a maximal torus $T$ and a B…

Let $T$) be a maximal torus in $\operatorname{GL}_n$. For $j=1,\dots,k$, let … be formal variables, and set $X^{(k+1)}=(T_1,\dots,T_n)$ to be the equivariant parameters in…

Let $H_r(\textbf{x})$ be the elliptic Ruijsenaars–Schneider Hamiltonians, with elliptic deformation parameter $\mathfrak{p}\in\mathbb{C}^\times$. Let $\mathcal{L}_{\textbf{d}}$ be…

Let $Fl=Fl(v_1,\dots,v_k;N)$ be a partial flag variety. Introduce variables $\gamma^i_j$ for $i=0,\dots,n$ and $j=1,\dots,\dim(\mathcal{R}_i)$, and let $Sym(\gamma)$ be the invaria…

Let $V//G$ be a GIT quotient satisfying the source's mild hypotheses, including that the $G$-unstable locus has codimension two. Let $T_G$ be a maximal torus of $G$, let $W$ be its…

Let $Fl$ be a partial flag variety. Denote by $QK(Fl)$ its stable-map quantum $K$-ring and by $QK^{QM}(Fl)$ its quasimap quantum $K$-ring. The quasimap–stable-map conjecture. In th…

Let $Fl=Fl(v_1,\dots,v_k;N)$ be a partial flag variety, let $W_Q$ be the discrete Wronskian matrix appearing in the Bethe-algebra description, and let $QK(Fl)$ denote its quantum…

Let $Fl$ be the partial flag variety and $T^*Fl$ its cotangent bundle. The quantum $K$-theory of $T^*Fl$ is considered together with the Bethe algebra of the Yang–Baxter algebra as…

Let $X=Fl(v_1,\dots,v_k;N)$ be the partial flag variety of flags $V_1\subset\dots\subset V_k\subset\mathbb{C}^N$ with $\dim(V_i)=v_i$. Let $\mathcal{S}_i$ be the tautological subbu…

Let $X=G/P$ be a flag variety over $\mathbb C$, let $W^{\mathrm{comin}}$ be the subgroup of point representatives of cominuscule flag varieties together with the identity, and let…

Let $X=G/P$ be a complex flag manifold. Let $w\in W^{\mathrm{comin}}$ and $u\in W$, let $e\in H_2(X,\mathbb Z)$ be effective, and let $M_{d(w,u)+e}(X_{w_0w},X^u)$ be the moduli spa…

Let $G$ be a reductive group acting on a vector space, let $T$ be a maximal torus with Weyl group $W$, and write $Y=V//T$ and $X=V//G$. Let $L_r$ be the bundle on $Y$ associated to…

Let $X=G/P_X$ be a rational projective homogeneous space. Schubert varieties in $X$ are indexed by $W^X$, and for $u\in W^X$ let $X^u$ be the Schubert variety with…