332 problems

For a smooth Fano variety $X$ of Hodge-Tate type, the modern form of Dubrovin's conjecture predicts … The refined Gamma conjecture II also identifies categorical and quantum data.…

Let $S$ be a smooth projective surface over the field of complex numbers ${\mathbb{C}}$. Consider its Hilbert scheme of points $\mathsf{Hilb}_n(S)$ and orbifold symmetric product…

Virasoro conjecture. For any smooth projective variety,

Let $X$ be a Fano manifold whose big quantum cohomology is semisimple, and suppose that $\mathcal{D}^b_{\rm coh}(X)$ admits a full exceptional collection. Let $U\subset B_{ss}$ be…

Let $X$ and $Y$ be birational Calabi-Yau manifolds, meaning smooth projective varieties with trivial first Chern class. Their quantum cohomology includes the big quantum cohomology…

Let $\mathrm{OG}$ be the odd quadric considered above, let $c_1=c_1(-K)\in H^2(\mathrm{OG})$ be its anticanonical class, and let $\operatorname{FPdim}(A(c_1))$ denote the Frobenius…

Fukaya-category elliptic-curve summand conjecture. The Fukaya category of $X$ should contain a direct summand equivalent to the derived category of sheaves on a genus-one curve who…

Frobenius structure conjecture. The coefficient of $\vartheta_0$ in the product

Let $(M,\omega)$ be a semipositive symplectic manifold, let $\phi:M\to M$ be a Hamiltonian diffeomorphism with finitely many periodic points, and let $\Lambda_{\mathbb{Q}}$ be the…

Let $(Y,D)$ be a smooth log Calabi–Yau orbifold with maximal boundary, meaning that $D$ contains a $0$-stratum and $D\in|-K_Y|$. Let $\operatorname{QH}_{\log}^0(Y^{\dagger})$ denot…

Givental's conjecture. The quantum $D$-module should be identified with the $S^1$-equivariant Floer theory of the free loop space.

Let $M$ be a projective manifold whose genus $0$ Gromov–Witten potential is semisimple, and let ${\mathscr D}_{Giv}$ be the formal total potential constructed from it by Givental's…

Let $\mathcal{X}$ be an orbifold with coarse moduli space $X$, and let $\phi\colon Z\to X$ be a crepant resolution. Iritani's conjecture. There exists an isomorphism … induced by a…

The cylindric Schur functions are symmetric functions that generate cylindric semistandard Young tableaux and arise in the study of the quantum cohomology of Grassmannians. A skew…

Completeness conjecture. The equivalences found in the study, including the zero and non-zero equivalences, form a complete collection of equivalences.

Galkin-Golyshev-Iritani's Conjecture O. Every Fano manifold $X$ satisfies: $\rho$ is an eigenvalue of $\hat c_1$ with multiplicity one; and for every eigenvalue $\lambda$ of…

Quantum Kirwan map conjecture. There exists a map

Let $(M,\omega)$ be a closed symplectic manifold and let $K\subset M$ be a compact subset. The subset $K$ is SH-heavy in the sense of the quantum cohomology ideal-valued quasi-meas…

Biran–Cornea's conjecture. Every monotone Lagrangian $L\subset\mathbb{C}P^n$ is either wide or narrow. This predicts a dichotomy for the quantum cohomology of monotone Lagrangians;…

Let $X$ be a Fano manifold, let $QH^*(X)$ be its quantum cohomology ring, and let $c_1$ denote the first Chern class of $X$. The quantum product by $c_1$ defines a linear operator…

Let $X$ and $Y$ be birational Calabi-Yau threefolds, meaning smooth projective varieties with trivial first Chern class. Morrison's conjecture. $X$ and $Y$ have identical small qua…

Monodromy conjecture. The monodromy of the quantum connection $\bigtriangledown$ for $X_i$ agrees with the monodromy induced on K-theory by the derived equivalence.

Intersection-cohomology–quantum-cohomology conjecture. The intersection cohomology of $X$ should be isomorphic to the quantum cohomology of $\tilde{X}$ specialized at $q=1$.

Canonical correspondence conjecture. There exists a canonical correspondence $T\in A(X\times X')$ that induces isomorphisms of cohomology groups and preserves the Poincaré pairing.

Let $X$ be a homogeneous space, and let $QH^*(X,\mathbb{Z})_{loc}$ denote its quantum cohomology ring localized by inverting the quantum parameter. The homogeneous space $X$ is cal…