32 problems

Let $g\geq 2$ be the genus, let ${\mathsf{r}}=2$ be the rank, and let ${\mathsf{d}}$ be the degree. Define the generating series $\check{\mathsf{QM}}^i_{\mathsf{d}}(q)$ and…

Let $G$ be the reductive group, $P$ a parabolic subgroup, and $\theta$ a degree, and let $\QM_{G,P}^{\theta}$ and $\mathcal{U}_{G,P}^{\theta_{\aff}}$ be the schemes introduced in t…

Let $\eta\in Y^+$ and suppose that $\eta-2\check\rho\in Y^+$. Let $K^\bullet_\eta$ be the graded multiplicity space of $\IC(\ol\fM_\eta)$ in $\bigoplus_{\xi\in Y}\bp_*\IC(\fQ^\xi)$…

Let $\fM$ be the moduli space of $\bG$-torsors on $\BP^1$ with formal trivialization at infinity, stratified by $\fM_\eta$ for $\eta\in Y^+$. Write … where $K^\bullet_\eta$ is the…

Let $\CF$ be a $\bG[[z]]$-equivariant perverse sheaf on the affine Grassmannian $\CG$. The natural projection … has target identified with … The algebra $\CA=U(\fn^L_+)$ acts on th…

Let $\CF$ be a $\bG[[z]]$-equivariant perverse sheaf on the affine Grassmannian $\CG$. For each $\beta\in Y$, let $T_{-\beta}$ denote the corresponding semiinfinite orbit, and cons…

Quasimap coefficient conjecture. There is an isomorphism of $\fg^L$-modules

Tilting conjecture. In the $\bG[[z]]$-equivariant case, this resulting $\fg^L$-module is tilting. This predicts that the geometric $\fg^L$-action on the global cohomology of quasim…

Let $G=SL_n$, let $\fn$ be the nilpotent subalgebra generated by the simple generators $e_1,\ldots,e_{n-1}$, and let $\CN\subset\frak{sl}_n$ be the nilpotent cone. For the quasifla…

Expected transversality conjecture. The intersection $I$ is generically smooth and pure of codimension

Let $X={\mathsf{Bl}}_{\mathbb{P}^s}(\mathbb{P}^r)$, let $C$ be a general curve of genus $g$, and let $\widetilde{Q}_\beta(C,X)$ be the moduli space of complete quasimaps $u:C\to X$…

Quantum module–Batyrev module conjecture. If $X$ is semipositive, then the quantum module $QM(X)$ is the same as the Batyrev module $\operatorname{BatM}(X)$.

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 $X=V\git T$ be a smooth semipositive toric variety, where $V$ is a finite-dimensional complex vector space and $T$ is a complex torus. Let $H^*(T)$ denote the group cohomology…

Fix cycles $c_i^X$ in the source $X=\mathbb{P}^k$ and cycles $c_i^Y$ in the target $Y=\mathbb{P}^n$, for $i=1,\ldots,l$. Let $\mathrm{Maps}_d(\mathbb{P}^k,\mathbb{P}^n)$ be the spa…

Let $C$ be a smooth curve of genus $g$, and let ${\mathsf r}$, ${\mathsf a}$, ${\mathsf d}$ and ${\mathsf w}$ denote the parameters in the quasimap invariant…

For a fixed rank ${\mathsf r}$, define the generating series obtained by summing over the flux index,…

Let ${\mathsf r}\geq 2$, and let $\check{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0}$ and $\hat{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0}$ denote the zero-degree quasimap invariants in…

Let $G$ be an abelian group acting in a CI-GIT presentation, and let $Y\mathord{/{\mkern-6mu/}}G$ be the corresponding quotient with projective coarse moduli space. Let…

Let $X$ be a toric variety and let $X^{\circ}$ denote its mirror. Consider the generating function of intersection numbers constructed as an analogue of Gromov–Witten invariants of…

The genus-one quasimap formula.

Let $Q^{\varepsilon}_{g,k}(Z(s),d)$ be the moduli space of stable quasimaps to the zero locus $Z(s)$, and let $LGQ^{\varepsilon}_{g,k}(E,d)'$ be the corresponding moduli space of L…

Ciocan-Fontanine–Kim wall-crossing conjecture.

Let $Z$ be a complete intersection in projective space, and fix $g,n\geq 0$. For each effective degree $\beta$, let $q^\beta$ be the corresponding Novikov monomial, let…

Let $E\sslash \textbf{G}$ be the relevant GIT quotient, let $I(\mathbf{t},z)$ be the quasimap I-function, let $P^Q(\mathbf{t},z)$ be the corresponding quasimap input, and let…