24 problems

Rosenberg's conjecture. 1. For $n>0$, the map

Blanc's lattice conjecture. If $\cC$ is smooth and proper over $\C$, then

Let $X$ be a projective variety. Holomorphic $K$-theory, denoted $K_{hol}^*(X)$, is obtained from spaces of holomorphic maps from $X$ to Grassmannians and loop groups; let $\beta$…

Karoubi's conjecture compares algebraic and topological $K$-theory for suitable locally convex or bornological algebras; here $(\mathcal{L}_1)_\infty$ is proposed as the replacemen…

Let $\mathcal{A}$ be a smooth $\mathbb{C}$-linear stable $\infty$-category. The finite Quillen-Lichtenbaum dimension conjecture. Then … Here $\dim_{\mathrm{QL}}\mathcal{A}$ is the…

Let $X$ be a smooth separated scheme over $\mathbb{C}$ of finite type and let $A$ be an abelian group. The semi-topological Quillen-Lichtenbaum conjecture. The canonical map … is a…

Let $K$ be a compact connected Lie group, $T$ its maximal torus, and $\pi\colon K\rightarrow K/T$ the natural projection. Denote by $G=K_{\mathbb C}$ the corresponding split reduct…

Let $\mathcal{C}$ be a $\mathbb{C}$-linear dg-category. Write $K_*^{\operatorname{top}}(\mathcal{C})$ for its topological $K$-theory groups and $HP_*(\mathcal{C})$ for its periodic…

Blanc's non-commutative de Rham comparison conjecture. This map is an equivalence. In some cases, the equivalence has been proved by A. Kahn. This is a non-commutative analogue of…

Higher units conjecture. For all $n\geq 0$ there are equivalences

The truncated Anderson dual conjecture. There is an equivalence of spectra

Let $X\to Y$ be a contraction of a smooth projective variety. Let $\operatorname{K}^{\mathrm{top}}(X)_\mathbb Q$ denote rational topological $K$-theory, equipped with its canonical…

K-theoretic computation conjecture.

Let $K^{-n}(X):=\widetilde{K}(\Sigma^n X_{+})$ for $n\geq 0$, where $\Sigma^n$ denotes the $n$-th reduced suspension and $K^0(X)$ is the Grothendieck group of structural vector bun…

The paper defines $e$ as a geometrically defined map from the $0$-connective algebraic $K$-theory space of the complex numbers to the homotopy fiber of the Chern character. Infinit…

Let $X$ be a compact real algebraic variety. For $4\mathbb{F}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}4$, let $d(\mathbb{F})=2^{e(\mathbb{F})}$, where … For a nonnegative integer $n$…

Let $\mathfrak A$ be a $C^*$-algebra, let $\mathcal K$ be the algebra of compact operators, and let $\mathfrak A\mathbin{\hat\otimes}\mathcal K$ denote their spatial tensor product…

Let $T$ be a smooth proper dg-category over $\mathbb{C}$. Let $\mathbf{K}^{\mathrm{top}}(T)$ denote its topological K-theory spectrum, $\mathrm{HP}(T)$ its periodic cyclic homology…

Let $A$ be a real or complex commutative Banach algebra with involution, let $\varepsilon$ denote the hermitian sign, and let $m$ be the coefficient modulus. The algebraic-to-topol…

Let $\mathcal R$ be a root datum with affine Weyl group $W^e$, let $\Gamma$ be the finite group acting in the extended affine Hecke algebra, and let $q$ be a positive parameter fun…

Let $\mathcal R$ be a root datum, let $W^e$ be its affine Weyl group, and let $\mathcal H(\mathcal R,q)$ be an affine Hecke algebra with positive parameter function $q$. Denote the…

Let $\Gamma<\operatorname{Isom}(\mathbb{R}^k)$ be a torsion-free crystallographic group. Its quotient $\mathbb{R}^k/\Gamma$ is a closed aspherical manifold, and…

The β-SrBC. The image of

Let $\beta$ be a bounding class, let $(G,L)$ be a discrete group equipped with a word-length, and let $H_{\beta,L}(G)$ be the associated rapid decay algebra. Write…