22 problems

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.

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

Non-spin extension conjecture. Theorem 3.1 holds even if $M$ is not spin; equivalently,

Let $X$ be a hyperbolic orbifold, and let $\lambda^{(0)}_1(X)$ denote the first non-zero eigenvalue of its Laplacian. Let $E_{\mathtt{spin}}\subset\mathbb{R}_{>0}$ be the set of th…

Gromov's conjecture. Then ${\rm \deg}(\Phi)=0$.

Let $g,n\geq 0$, let $k>0$, and let $a\in\mathbb{Z}^n$ satisfy $|a|=k(2g-2+n)$, $a\notin k\mathbb{Z}_{>0}^n$, and the condition that all entries of $a$ and $k$ are odd. For odd sta…

Let $\mathcal{M}_g(2g-1)$ be the minimal stratum of differentials, let $\chi(\mathcal{M}_g(2g-1))^{\rm spin}$ denote its spin Euler characteristic, and let…

Let $(X,g)$ be an $n$-dimensional compact connected orientable manifold with boundary, and let $\underline{X}_\bullet$ be a closed orientable manifold of dimension $n-m$. Suppose ……

Let $M$ be a closed spin manifold. Its Rosenberg index is the index class associated with the spin Dirac operator, and $M$ is said to have infinite $\mathcal{KO}$-width when its…

Let $p$ be a nonnegative integer and $q$ a natural number. The form … can be realized as the intersection form of a closed smooth spin $4$-manifold. Matsumoto's 11/8-conjecture. Th…

Gromov–Lawson conjecture. The higher $\widehat A$-genus vanishes.

Let two three-dimensional lens spaces be ε-isospectral if their Dirac spectra agree up to the equivalence encoded by ε, and let them be ε-spin-isometric if there is an ε-compatible…

Non-existence conjecture. Two Dirac isospectral spin lens spaces of dimension $n\equiv 1\pmod4$ are necessarily isometric.

Moduli-group rank conjecture. The rank of $\widetilde P(M)$ is at least $N_{\mathrm{fin}}(\Gamma)$.

Local extension conjecture. There is a unique parabolic monogenic spinor $\Psi$ converging on a neighborhood of $\mathcal V$ in $\mathcal U$ such that

Main conjecture. The following conditions are equivalent: $(M,\phi,s)$ and $(M',\phi',s')$ are $Y$-equivalent; and there is a homology isomorphism from $(M,\phi)$ to $(M',\phi')$ s…

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

Let $M$ be a connected Riemannian spin manifold, and set … Let $\phi$ be a nonzero spinor satisfying … for some Riemannian metric $g_0$ and $c\in\mathbb{R}$. Generic nonvanishing c…

Let $M$ be a globally hyperbolic spin spacetime and let $O\subset\mathcal{M}$ be a bounded cc-region with non-empty causal complement. Let ${\mathbf{F}}^0$ and ${\mathbf{R}}^0$ den…

Let $G_{2k,2\ell}=SO(2(k+\ell))/(SO(2k)\times SO(2\ell))$ be the even-dimensional spin Grassmannian, and let $\lambda$ denote a weight occurring in the spinor-bundle decomposition.…

Let $G_{2k,2\ell}=SO(2k+2\ell)/(SO(2k)\times SO(2\ell))$ be the even-dimensional spin Grassmannian, and let the spinor-bundle decomposition be the one described in the preceding de…