16 problems

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 $M_\varnothing\simeq\mathsf{Gr}^+(2,5)\simeq\mathsf{Sp}(2)/\mathsf{U}(2)$ be the nodal manifold, identified with the complex quadric threefold $Q^3$, and suppose its nontrivial…

Let $M_\medcirc\simeq\mathbb{C}P^3$ be the nodal manifold in the considered class-$\textrm{D}$ model, with cells in dimensions $D\in\{0,2,4,6\}$. Chirally resolved Chern-number con…

Let $M_\medcirc$ be the $11$-dimensional nodal manifold in the quaternionic chiral symmetry class $\textrm{CII}$, and let the enclosing sphere be $S^{15}$. The manifold contains a…

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…

Let $M$ denote the matrix size, and consider the universal Chern-number distribution arising in the random matrix field model. Asymptotic universality conjecture. The universal dis…

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 $\mathbb{C}P^n$ be complex projective $n$-space, and let $S$ be a two-dimensional submanifold of $\mathbb{C}P^n$ with vanishing scalar curvature. Suppose that $S$ is a torus an…

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 $H_\omega$ be a covariant Hamiltonian and let $\mu$ be the Fermi level. Its Fermi projection is … Consider topological phases in even dimensions whose classification is by…

Maximal Fano degree conjecture. Among all $n$-dimensional Fano manifolds $M$, the maximum of $c_1^n(M)$ can only be attained by $\mathbb{P}^n$.

For rational magnetic flux $p/q$, let $r$ denote the gap label and let $\sigma_{\rm H}$ be the Chern number associated with the gap, satisfying the Diophantine equation … for some…

Generic sign-ambiguity conjecture. For generic Hofstadter models, the mod $q$ ambiguity in the solution of the Diophantine equation is a sign ambiguity: the Chern number is either…

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…