7 problems

Signed Euler characteristic conjecture. Under these hypotheses,

Let $M$ be a simple matroid of rank $d+1$ on a ground set $E$ of size $n$. For nonnegative integers $k_1,\ldots,k_d$ with $k_1+2k_2+\cdots+dk_d=d$, consider the Chern number…

Let $X_n \to Y_n \to (Z,D)$ be a terminal good partial resolution of singularities of the $n$-th root cover construction. Assume that $K_{Y_n}$ is nef, and let $X'_n$ be a minimal…

The known irreducible hyper-Kähler manifolds include Hilbert schemes of points on K3 surfaces, generalized Kummer varieties, and two further examples in dimensions $6$ and $10$. A…

Let $(M,\omega)$ be an $n$-dimensional compact Kähler manifold with nonpositive holomorphic bisectional curvature, and suppose its Ricci curvature is quasi-negative, meaning nonpos…

Let a two-dimensional photonic crystal have a boundary and bulk band gaps. In each bulk band gap, consider the left- and right-moving boundary modes, and let the bulk Chern number…

Let $X$ be a Kähler–Einstein toric Fano $n$-fold, and let $H$ be a nef class on $X$. Chan–Leung's conjecture. … Chan and Leung proved this in dimensions $n=2,3,4$ and when every fa…