18 problems

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 $M$ be a compact strictly pseudoconvex Cauchy-Riemann $3$-manifold and let $\theta$ be a contact form on $M$. Write $R$ and $\tau$ for the Tanaka-Webster curvature and torsion…

Let $q>1$ and let $(L,\varphi)$ be an $F_m$-link concordance class in $C_{2q+1}(F_m)$. For each positive integer $k$, let $R_k(F_m)$ denote the space of $k$-dimensional unitary rep…

Let $N\to S^1$ be a fibration with a family $D$ of twisted Dirac operators on the fibers, and let $\delta_t$ be the associated self-adjoint adiabatic family of operators for $t>0$.…

Let $X$ be an odd-dimensional manifold with the geometric data used to define the eta-type invariant, let $g$ be the corresponding gauge transformation, and let $D^{[0,1]}$ be the…

Full twisted eta-invariant conjecture. The full set of twisted $\eta$-invariants of the Dirac operator modulo $1$, collected into this function on the set of orbi-bundles…

Let $S^5/{\mathbb Z}_N$ be the boundary of the orbifold construction, and let $\beta$, $\gamma$, $\widetilde{\delta}_k$, and $\widetilde{\epsilon}_k$ denote the anomaly coefficient…

Let $A$ be the asymmetry operator associated with the massless Dirac operator $W$ on $M$, and let $\psi_\mathrm{Dir}^\mathrm{loc}(x)$ and $\psi_\mathrm{Dir}$ denote its regularised…

Let $H$ be a group, let $h\in H$, and let $D_P$ and $D_{\mathcal P_M}$ be the operators associated with the boundary conditions $P$ and ${\mathcal P}_M$. Let $A$ be the relevant bo…

Let $\mathcal{A}_0$ and $\mathcal{A}_1$ be the self-adjoint strongly Callias-type operators for which the relative eta function $\eta(s;\mathcal{A}_1,\mathcal{A}_0)$ is defined. As…

Let $alpha ^0$ and $alpha ^1$ be the invertible field theories whose partition functions are the exponentiated eta-invariants formed from the operators $D^0$ and $D^1$, respectivel…

Let $\widetilde{D}_M$ be an $L^2$-invertible $\Gamma$-equivariant Dirac operator on a Galois $\Gamma$-covering $\widetilde{M}\to M$, acting on sections of a $\Gamma$-equivariant co…

Let $M$ be a non-compact manifold with boundary, and let $\mathcal{A}_0$ and $\mathcal{A}_1$ be the boundary operators for the relevant strongly Callias-type operators. Suppose tha…

Let $X$ be an even-dimensional manifold, let $E$ be the relevant flat Hermitian vector bundle with associated Dirac operator, let $g$ be the unitary automorphism used to define the…

Let $Y$ be the Milnor fibre compactification equipped with the product-type metric described above, and let $D$ be its signature operator. Denote by $D_{I-P_{Y_0}}$ the operator wi…

Counterterm formula conjecture. The counterterm in the preceding topological-invariance conjecture is

Let $(X,\phi,\xi,\kappa,g)$ be a closed quasi-regular K-contact three-manifold, where $g=\kappa\otimes\kappa+d\kappa(\cdot,J\cdot)$. Let $D$ be the contact operator, and let…

Differential equivariant pushforward conjecture. The equality holds as class functions modulo $R(\Gamma)$. This conjectures that the index-theoretic description of the ordinary equ…