12 problems

Let $[\Sigma]$ be an integral homology class, with reductions $[\Sigma]^{\mathbb{R}}$ and $[\Sigma]^{\mathbb{Z}/n\mathbb{Z}}$ in the corresponding coefficient systems. The Hasse pr…

Consider the direct-limit coefficient group $\mathbb{Q}/\mathbb{Z}$ obtained from the groups $\mathbb{Z}/n\mathbb{Z}$ under the multiplication maps, equipped with the norm induced…

Let $p$ be a prime. Consider a variational problem formulated on chain spaces over $\mathbb{Q}_p$, $\mathbb{Z}_p$, and $\mathbb{Z}/p^n\mathbb{Z}$ for $n\geq 1$, all equipped with t…

Let $d$ and $c$ be positive integers, and let $T$ be a $d$-dimensional $\mathbb{Z}$-area-minimizing integral current in $\mathbb{R}^{d+c}$. Assume that $T$ has Euclidean volume gro…

A variational problem may be formulated on chain spaces with coefficients in $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Z}/n\mathbb{Z}$ for $n\geq 2$. Obstructions to the Hasse princ…

A variational problem may be formulated on chain spaces with coefficients in $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Z}/n\mathbb{Z}$ for $n\geq 2$. The Hasse principle for variati…

Let $k\in\mathbb{N}$, let $\mathrm{X}$ be a metric space, and let $\mathbf{M}_k(\mathrm{X})$ denote the space of metric $k$-currents on $\mathrm{X}$. Normal $k$-currents are metric…

Lang's general Flat Chain Conjecture. For every $d\in\mathbb{N}$ and $k\in\mathbb{N}\cup\{0\}$, if $T\in\mathcal{D}_k(\boldsymbol{R}^d)$ has compact support, then

Finite-flat-norm variant of Lang's conjecture. Every compactly supported metric $k$-current in $\boldsymbol{R}^d$ with finite flat norm is a flat chain.

Top-dimensional Flat Chain Conjecture. There exists some $u\in L^1(Q)$ such that

Lang's Flat Chain Conjecture. Every compactly supported metric $k$-current in $\boldsymbol{R}^d$ is a flat $k$-chain.

Let $T$ be a metric $k$-current on Euclidean space, and let $\widetilde{T}$ denote the associated flat chain. The map sending a flat chain with finite mass to the metric $k$-curren…