10 problems

Homological mirror symmetry conjecture. The wrapped Fukaya category of $U$ should be equivalent to the bounded derived category of coherent sheaves on its corresponding mirror:

Theta-function multiplication conjecture. The multiplication rule of theta functions should have a combinatorial description in terms of broken lines defined in the work of Gross,…

Let $D^{\perp}\simeq\mathbf{D}_{2m}$ with $m\geq 2$, let $1\leq l\leq m-1$, and let $\mathcal{P}$ be the power set of $\{1,\cdots,l\}$. For each…

The resulting invariants are the log BPS invariants $m^P_\beta$ and $m^{tot}_\beta$ associated with a log Calabi-Yau surface $(S,E)$, while $\operatorname{Tot}(K_S)$ denotes the to…

Let $(S,E)$ be a log Calabi-Yau surface with respect to a class $\beta\in\operatorname{H}_2(S,\mathbb{Z})$. The total log BPS number $m^{tot}_\beta$ is defined by the multiple-cove…

Let $B$ be the base of the log Calabi–Yau surface, and let $\beta\in\operatorname{NE}(Y)$. Let $\gamma$ be a broken line in $B$ for the canonical scattering diagram, and choose a n…

Let $A$ be the algebra with theta-function basis $(\vartheta_p)$ and trace pairing $\operatorname{Tr}(f,g)$, and write $\vartheta_q^{-1}$ for the inverse of $\vartheta_q$ when defi…

Let $\mathcal X/S'$ be the mirror family, let ${S'}^o$ be its smooth-fiber locus, let $\gamma_s$ be the torus-fiber class, and identify…

Let $U=Y\setminus D$ and let $X$ be the mirror fiber of the constructed family over $s=\exp(2\pi i(\mathbf B+i\omega))$. Symplectic-cohomology mirror conjecture. There is an isomor…

Let $Y$ be a smooth projective surface, let $D$ be an anticanonical divisor supporting an ample divisor, let $R=\Bbbk[\operatorname{NE}(Y)]$, and let…