12 problems

Hesselholt's descent conjecture. For all positive integers $q$, this canonical map is an isomorphism of pro-abelian groups, and the higher continuous cohomology groups of the pro-…

Let $K$ be a field with valuation ring $V$, residue field $k$, and let $\bar{K}$ be an algebraic closure of $K$ with Galois group $G_K$. Let $\bar{V}$ be the integral closure of…

Let $\bbS^{\widehat{\calN}}$ be the stack of one-dimensional commutative formal groups with a section, and let…

Goodwillie's conjecture. The source introduces $K^{\mathrm{ind}}$ as the fiber of this trace map.

Quasi-isomorphism conjecture. The second pages of these two spectral sequences are quasi-isomorphic; equivalently, the subcomplex consisting of terms with non-integer exponents is…

At an odd prime $p$, let $\Fib(\tau)$ denote the fiber of the cyclotomic trace for the sphere spectrum, and let $I_{\mathbb{Z}_p}$ denote the $\mathbb{Z}_p$-Anderson dual. The fibe…

Let $X$ be a nice scheme over $\mathbf{F}_{p}$, let $K(X)$ denote its algebraic $K$-theory spectrum, and let $TC(X,p)$ denote its $p$-typical topological cyclic homology. Cyclotomi…

Let $R$ be a connective $E_{\infty}$ ring spectrum, and let … be the cyclotomic trace, where both spectra are $p$-completed. The cyclotomic-trace compatibility conjecture. The cycl…

Let $\mathbb{F}$ be a field, let $A=H\mathbb{F}$, and let $R$ be the algebra in the spectral sequence described above. Its $E^{2}$-term is the second page of the spectral sequence…

Let $\operatorname{TR}^3_{13-\lambda_4}(\mathbb{Z};2)$ be the indicated equivariant topological Hochschild homology group, where $\lambda_4$ is the relevant $RO(S^1)$-representatio…

Let $\operatorname{TR}^3_{9-\lambda_4}(\mathbb{Z};2)$ be the indicated equivariant topological Hochschild homology group, where $\lambda_4$ is the relevant $RO(S^1)$-representation…

Let $T(\mathbb{Z})$ denote the topological Hochschild homology spectrum of $\mathbb{Z}$, and consider the $C_4$-Tate spectral sequence converging to the homotopy groups of…