40 problems

Let $X$ be a smooth scheme. Consider the non-linear $\mathcal{D}$-Fredholm index, its symbol complex, and the cycle defined by the associated graded symbol morphism in the total sp…

Uniform geometric K-homology conjecture. The natural map

Relative index comparison conjecture. There exists a map

Let $N\to B$ be a fiber bundle with base $B$ a connected closed manifold, and with all fibers diffeomorphic to a fixed connected closed manifold $Z$. Suppose there is no fiberwise…

Let $(M,\sigma)$ be a closed connected totally non-spin spin$^c$ manifold with $\dim M=n$, fundamental group $\pi$, and classifying map … Let $L$ be the associated spin$^c$ line bu…

Novikov's conjecture. The higher signatures are oriented homotopy invariant. The conjecture is a central problem in index theory and topology. In the paper's setting, it is related…

Strong Novikov conjecture. The induced map

Coarse Novikov conjecture. The induced map

Let $(M,L)$ be a closed spin$^c$ non-spin manifold with $\dim M\geq 5$ and with $\mathcal R^{{\mathrm{gen}+}}(M,L)\neq\emptyset$. Fix a basepoint…

The coarse assembly injectivity conjecture. The map $\mu$ is injective. Injectivity of the coarse assembly map is the coarse Novikov property; the paper establishes this property f…

Let $\mathrm{SingMet}$ and $\mathrm{ConcSet}$ be the cubical models of, respectively, the space of positive scalar curvature metrics and the space of positive scalar curvature conc…

Let $H_{\mu,\nu}$ be the operator parametrized by $\mu,\nu\geq0$, let $H_{\rm D}$ be the Dirichlet reference operator, and let $W_-(H_{\mu,\nu},H_{\rm D})$ denote the incoming wave…

Let $k$ and $l$ be relatively prime integers with $k>l>0$, and let $\widetilde{IFS}_{k,l}$ denote the orientable cover of the Fraser-Sargent free-boundary minimal surface…

Let $k$ and $l$ be relatively prime integers with $k>l>0$, and let $EFS_{k,l}$ be the exterior Fraser-Sargent surface in $\mathbb E^4\setminus\mathring{\mathbb B}^4$ associated wit…

Unstable relative Gromov–Lawson–Rosenberg conjecture. If the relative higher index of $D$ is zero in

Gromov's compactness conjecture. If the $\sideset{}{^1}\varprojlim$ higher index of the Dirac operator of $X$ vanishes, then $X$ admits a complete Riemannian metric with scalar cur…

Let $E_{b4}(Z)$ be the operator associated with the elliptic complex on the periodic end, and let $\widehat{H}^*_z(V)$ denote the corresponding periodic cohomology group, with para…

KO-width–Rosenberg index conjecture. One has

Stable Gromov–Lawson–Rosenberg conjecture. A spin manifold stably admits a positive scalar curvature metric if and only if

Let $\mathcal F$ be an $\mathcal N=(0,1)$ SQFT with cylindrical ends and boundary $\partial\mathcal F$, satisfying $Z_{RR}(\partial\mathcal F)=0$, and suppose $\partial\mathcal F$…

Let $X_{\Sigma}$ be a spin manifold with a $(\eta\text{-}\mathrm{fb})$-singularity of dimension $n\geq 9$. Assume that $\beta X\neq\varnothing$ and that…

Generalized Stolz conjecture. The index map $\Theta$ is an isomorphism. This is the proper-action analogue of the Stolz conjecture, replacing $B\Gamma$ by the universal proper…

Let $(L,f,h)$ represent an element of $R^{\textup{spin}}_n(B\Gamma)$, where $L$ is an $n$-dimensional spin manifold with boundary carrying a positive-scalar-curvature metric $h$, t…

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 … , let … be the APS $e$-invariant map. The notation $[E,F_{t,}]$ and $[B,F_{t,}]$ denotes homotopy classes of maps into the indicated target. Finer APS index conjecture. The f…