10 problems

Douglas's conjecture. Conjecture A holds true, and its induced extension class $\tau_I$ is identified with the fundamental class of $X_I$, namely the extension class induced by the…

Let $M$ be a compact oriented smooth manifold with smooth boundary $\partial M$, let $E_0$ and $E_1$ be smooth complex Hermitian vector bundles over $M$, and let…

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$ with smooth boundary, let $\mathcal{Z}$ be a subvariety, and consider the odd $K$-homology class determine…

Let ${\mathcal{S}}$ be an essentially reductive submodule of ${\mathcal{M}}_\Lambda\otimes {\mathbb{C}}^k$, and let $[{\mathcal{S}}^\bot]$ denote the associated element of…

Let $Sp(n)$ be the compact symplectic group, let $U(i)$ be the unitary group, and let $K.\Omega Sp(n)$ denote the K-homology ring of its based loop space. For $1\leq i\leq n$, the…

Let $X$ be a compact space. A generalized elliptic operator over $X$ is a triple $(D,H_1,H_2)$, where $D:H_1\longrightarrow H_2$ is a Fredholm operator on Hilbert spaces $H_1$ and…

Uniform geometric K-homology conjecture. The natural map

Equivariant Baum–Douglas conjecture. Equivariant analytic and geometric $K$-homologies should be equivalent in general; equivalently, the map $BD$ should be an isomorphism.

Let $p\in\mathbb{C}[z_1,\ldots,z_n]$ be a polynomial, let $X_p$ be the associated compact subset of $\partial\mathbb{B}_n$, and let $[p]\in K_1(X_p)$ be the odd K-homology class de…

Let $SG$ denote the Sierpiński gasket and let $(A,H,D)$ be a spectral triple for $C(SG)$ inducing the geodesic distance on the gasket. Suppose that it is summable for every…