10 problems

Let $X$ be a compact Riemann surface, let $P\subset X$ be a finite set, and let $(V,\nabla_V)$ be a semisimple holomorphic flat bundle on $X\setminus P$. Let $(H,D_V,h)$ be the tam…

Let $(A)_d$, $(B)_d$, $(C)_d$, and $(D)_d$ be hexagonal lattice diagrams, and let the corresponding bridge trisections be those determined by these diagrams. Deperturbation conject…

Let $X$ be a Kähler manifold or a Kähler orbifold with first Betti number $b_1=0$ and second Betti number $b_2\geq 2$. Suppose that $X$ contains disjoint complex curves spanning…

Unitary representation conjecture. Representations of this group approximate, as $N\to+\infty$, complex curves in the abelian variety $(\mathbb{C}/(\mathbb{Z}+i\mathbb{Z}))^2$. Thi…

Efficient bridge trisection conjecture. If

A transverse $\mathbb{C}$-link in $S^3$ is a link arising as the transverse intersection of a smooth complex curve with the unit sphere in $\mathbb{C}^2$. Rudolph's conjecture. Eve…

Nonexistence conjecture. There does not exist such an $X$ containing $b_2(X)$ disjoint complex curves all of genus $g\geq 1$.

Curve critical-point conjecture. For every $k$, the functional $\mathcal{F}_{k}$ admits a critical point.

Fang's conjecture. The upper bound is achieved precisely by the Fubini–Study metric on $L$, up to scaling.

Gillet–Soulé's conjecture. The determinant of $\Delta_{\bar{\partial}}$, viewed as a functional on the space of all smooth Hermitian metrics on $L$, is bounded from above.