87 problems

Gromov–Lawson–Rosenberg conjecture. The manifold $M$ admits a psc-metric if and only if its Rosenberg index vanishes.

Let $M$ be a spin manifold with higher $alpha$-invariant, and let $\mathrm{Bott}$ denote the Bott manifold. Stable Gromov-Lawson-Rosenberg conjecture. If the higher $alpha$-invaria…

Let $M$ be a smooth spin manifold, and let $\alpha(M)\in \operatorname{KO}_m(\mathbb{C}^*\pi_1(M))$ denote its Rosenberg index, where $m=\dim M$. Rosenberg's universal index obstru…

Let $N$ be a closed manifold of dimension at least $5$. The Rosenberg index is the index-theoretic obstruction to positive scalar curvature obtained from the Dirac operator with co…

Schroeder–Strake conjecture. Under these assumptions, $(M^n,g)$ is flat.

Conjecture 6.1. If $\frac{1}{2}p_1(F)=0$ and the Ricci curvature of $g^F$ along each leaf is positive, then the Witten genus

Let $M$ be a closed Riemannian spin manifold of dimension $n\ge 3$, let $S$ denote its scalar curvature, and let $\lambda$ be an eigenvalue of its Dirac operator. The Dirac eigenva…

Let $V=V^n(a_1,\dots,a_r)$ be a smooth complete intersection of complex dimension $n$ in $\mathbb{C}P^{n+r}$, defined by hypersurfaces of degrees $a_1,\dots,a_r$, and let $g$ be a…

Extension conjecture. The formula of Theorem 2.5 should also hold for any even-dimensional compact spin manifold with boundary.

Let $M^m$ be a closed spin manifold with fundamental group $\pi$, and let $u:M\to B\pi$ classify its universal covering. The index class is an element … A group is torsion free whe…

Arbitrary finite Dirac-spectrum prescription conjecture. Apart from these algebraic constraints and the index-theoretic constraints on the zero eigenvalue, there is a Riemannian me…

Generic-metric conjecture. For a generic Riemannian metric on any compact spin manifold, the space of harmonic spinors is no larger than forced by the index theorem.

Generic metrics conjecture. For generic metrics on $M$, the dimension of the space of harmonic spinors is the minimal dimension prescribed by index theory.

Existence conjecture. Every spin manifold $M$ admits a metric with harmonic spinors.

Unduloid-extremizer conjecture. In the case $x=0$ and $y<1$, the maximizers and minimizers correspond to the unduloid immersions.

Cylinder-extremizer conjecture. These solutions are exactly those corresponding to the maximizers and minimizers of $\mathcal{F}_{q_D}$, when all spinors are normalized to have…

Nodal-set dimension conjecture. The nodal set of $\varphi$ has Hausdorff dimension at most $n-2$.

Let $M$ be a compact connected spin manifold, let $D_g$ be the Dirac operator for a Riemannian metric $g$, and call $g$ $D$-minimal when the kernel of $D_g$ is no larger than force…

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 $X$ be a compact smooth spin manifold. Consider the elliptic operators obtained from the formal twisted Dirac operators … Here $D$ is the Dirac operator and $q$ is a parameter.…

Realisability conjecture. The bundles $Q$ and $Q'$ are equivalent if and only if $\varphi$ is realisable.

Monotonicity conjecture. The quasi-local mass $m(\Sigma)$ has increasing monotonicity.

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…

The 11/8 conjecture. The bound

Let $M$ be a spin manifold with finite fundamental group of odd order, let $\widetilde M$ be its universal cover, and let $\alpha(\widetilde M)$ and $\alpha(M,u)$ denote the corres…