12 problems

Let the spectral curve be … Here the stationary Gromov–Witten theory of ${\mathbb P}^1$ refers to Gromov–Witten invariants with stationary insertions, and topological recursion is…

Givental's conjecture. The generating function $G(\{T_k^+\},\{0\})$ is equal to one, or equivalently

Let the stationary Gromov–Witten invariants of $\mathbb{P}^1$ be the Gromov–Witten invariants with stationary insertions, and let TR denote topological recursion applied to the rel…

Let $\mathcal{J}$ be the Jouanolou device of $\mathbb{P}^1$, and let $\mathbf{G}_0\subseteq\mathbf{G}$ be the subgroups introduced from the explicit generators, with…

Let $\mathcal{J}$ be the Jouanolou device of $\mathbb{P}^1$, and let $[\mathcal{J},\mathbb{P}^1]^\mathrm{N}$ and $[\mathbb{P}^1,\mathbb{P}^1]^{\mathbb{A}^1}$ denote the correspondi…

Let $G(\alpha;n,d,k)$ denote the moduli space of $\alpha$-stable coherent systems of type $(n,d,k)$ on the projective line. Suppose that $a\ge2$, $1\le t\le n-1$, and $k\ge n$. Mon…

Non-emptiness conjecture. The space $G(\alpha;n,an-t,k)$ is non-empty for some $\alpha>0$ if and only if one of the following holds:

Pandharipande's degree-one formula. The degree-one stationary Gromov–Witten invariant of $\mathbb{P}^1$ satisfies

Let $X_n$ denote the locus of configurations $(P_1,\ldots,P_n)$ for which the canonical ring has exceptional generation behavior, and let the fat diagonal be the locus where at lea…

Let $D$ be a … , supported at points $P_1,,P_n$, and let $S_D$ be its canonical ring. A minimal relation means a relation in a minimal presentation of $S_D$ by generators. The poin…

Let $K$ be a function field over a field $k$, and let $\varphi$ be a self map of degree $d$ of $\mathbb{P}^1_K$. Let $R(\varphi)$ be the minimal resultant and let…

Let $\varphi$ be a self map of the projective line and let $\Phi$ be a semi-stable presentation of $\varphi$. A presentation is called minimal when it is minimal at every place. Mi…