107 problems

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…

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

Coarse Novikov conjecture. The induced map

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…

Gromov's band-width conjecture. The metric $g$ satisfies

Let $X$ be an $n$-dimensional Riemannian manifold. For each $q$ with $0\leq q\leq n$, let $\Delta^{(q)}$ be the Laplace operator on differential $q$-forms, let $h_t^{(q)}$ be its a…

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 $M$ be a uniformly contractible Riemannian manifold with bounded geometry. Gromov's conjecture. $M$ cannot have uniformly positive scalar curvature. The conjecture is presented…

Let $H$ be the self-adjoint operator associated with the model, and let $\hat\Gamma$ be its grading operator. Define … and let $H_+:=H|_{{\cal H}_+}$, viewed as an operator from…

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…

Let $K$ be a totally real number field, let $X$ be the associated Hilbert modular space with cusp singularities, and let a cusp singularity have a signature defect. Let the Shimizu…

Let $A$ be a self-adjoint elliptic operator of odd order on an even-dimensional manifold, and let its eta-invariant be defined in the usual sense. Gilkey's conjecture. The eta-inva…

Let $Y$ be a contact manifold, and let $X_1$ and $X_2$ be nonsingular Stein fillings of $Y$. Let $D^+_1$ and $D^+_2$ be the Dolbeault-Dirac operators on the fillings, and define th…

The gluing index conjecture. The expression above plays the role of the index of the object $X$ obtained by gluing $X_1$ and $X_2$ along $Y$, and hence is equal to the relative ind…

Let $Y$ be a contact manifold, let $F$ be a vector bundle over $Y$, and let ${\cal J}_1$ and ${\cal J}_2$ be polarizations corresponding to fillings $X_1$ and $X_2$. Assume that…

Let $Y$ be a contact manifold with embeddable CR structures $J_1$ and $J_2$, and let $X_{J_1}$ and $X_{J_2}$ be their nonsingular Stein fillings. Glue these fillings along their co…

Let $T_t$ be the family of operators described above, let $S$ be the associated selfadjoint operator, and suppose that $0\in\operatorname{spec}S$. For small $t$, the selfadjoint ex…

Let $(Y,H)$ be a compact, 3-dimensional contact manifold, and let $\mathcal{S}_0$ and $\mathcal{S}_1$ be Szegő projectors associated with embeddable, strictly pseudoconvex CR-struc…

Let $D$ be a generalized Dirac operator along the leaves of a foliation with Hausdorff graph, and suppose that the projection onto the kernel of $D$ and the spectral projections of…

Moy–Yakimova index conjecture. One has

Let $Gamma$ be a discrete group, let $BGamma$ be its classifying space, and let $mu$ be the assembly map. The rational assembly map is … The Strong Novikov conjecture. The map…

Index formula conjecture. For all subsets $S$ and $T$ of $\Pi$,

Tauvel–Yu index formula conjecture. The upper bound is in fact an exact formula:

Let $M\subseteq H^2\otimes\mathbb C^r$ be generated by vector polynomials, let $\bar T$ be the finite-rank $d$-contraction obtained by compressing the rank-$r$ $d$-shift to…

Let $H$ be a manifold with corners and let $D_H$ be a cusp pseudodifferential operator of order $1$ that is of Dirac type near the boundary, as in the referenced theorem and remark…