11 problems

Let $M$ be a compact spin manifold of dimension at least three, and let $h_g$ denote the dimension of the space of harmonic spinors for a Riemannian metric $g$ on $M$. Harmonic-spi…

Let $M$ be a closed spin surface. A harmonic spinor is a nonzero element of the kernel of the Dirac operator for a Riemannian metric and spin structure on $M$. The uniqueness conje…

Corrected Hitchin conjecture. For a generic complex structure on $X$,

Hitchin's conjecture. For a generic complex structure on $X$,

Inequivalent indefinite metrics conjecture. For a given Lie algebra $\mathfrak g$, there are a number of inequivalent indefinite metrics on $\mathfrak g$.

Let $\mathcal P$ be the parameter space for the Seiberg–Witten equations, let $\mathcal M_{SW}$ be their moduli space of solutions, and let…

Let $Y$ be a 3-manifold, let $p=(g,B)\in\mathcal P$ be a parameter pair with smooth metric $g$, and let $(\mathcal Z,\ell,\Phi)$ be an element of the moduli space of $\mathbb Z_2$-…

Genericity conjecture. Assumption 1 also holds generically within the set of pairs $(g_0,B_0)$ that admit $\mathbb{Z}_2$-harmonic spinors.

Let $M$ be a three-dimensional manifold with metric $g$, and let $\Sigma$ be the zero locus of a $\mathbb{Z}/2$-harmonic spinor, so that $\Sigma$ is a closed set of Hausdorff dimen…

Let $M$ be a closed spin manifold of dimension $n\geq 3$, let $R(M)$ denote its space of Riemannian metrics, and for $g\in R(M)$ let $D_g$ be the associated Dirac operator on the s…

Harmonic-spinor existence conjecture. There exists a Riemannian metric $g$ on $M$ such that