14 problems

Let $M$ be a smooth manifold, let $D(U,M)_2$ denote the relevant smooth $2$-simplices over $U$, and let $D(\mathbb{A}^n,D(U,M)_2)_1$ denote the corresponding smooth $1$-simplices.…

Let $(M_{8\ell+2},g_M^{\mathrm{spin}})$ be a closed differential spin manifold, and let $g_M^{\mathrm{string}}$ be a lift of its spin structure to a differential string structure.…

Let $H$ be a compact, simple and simply connected Lie group, and let $\mathcal{G}_{\mathit{bas}}$ be its basic gerbe. Write … for the extension of smooth group objects in…

Let $T_3$ be the restriction to the 1-cycle $\gamma$ of the S-duality torus bundle, and let $L_\gamma^t$ denote the pairing of the relevant torsional twisted classes. For monodromy…

Hopkins's block upper-triangular characteristic-class conjecture. This induced map is an isomorphism. The statement appears among results attributed to ideas of Bott, but the sourc…

Let $F$ be a sheaf on the category of smooth manifolds, and let $\mathsf{Cor}_{\mathrm{fcov}}(\mathsf{Mfld})$ be the 2-category whose morphisms are correspondences with finite cove…

Vanishing conjecture. The groups vanish for all $k$:

Let $\Gamma$ be a pseudogroup. Consider the universal map … and the characteristic map … associated with an almost $\Gamma$-structure $P$ on a manifold $M$. The plan is to find cla…

Let $M$ be a smooth manifold. Write $4\pi_0(\mathrm{FFT}_2^{sf}(M)^\times)4$ for the set of isomorphism classes of two-dimensional superficial, invertible smooth functorial field t…

Let $H_n$ be the symmetry group in dimension $n$, let $\operatorname{Bord}_n^{\nabla}(H_n)$ be the geometric bordism category with connections, and let $F$ be an invertible theory…

Let $X$ be a closed Riemannian manifold of odd dimension $d$ equipped with a generalized Dirac operator $\slashed{D}$. Let…

Transfer index conjecture. The two maps of $\Bundle^{\delta}_{geom}$-sets are equal:

Let $\pi:M\to B$ be a proper submersion between manifolds, let $V$ be a bundle of $R$-modules with geometry $h^{V}$, let $g^{\pi}$ be a Riemannian structure on $\pi$, and let…

Let $\mathcal{A}^e$ be the algebra whose linearity defines the relevant class of maps, let $t$ be the homotopy map referred to in the source, and let $\overline{t_k}$ denote the pr…