The Lipschitz and $C^1$ harmonic capacities $\mathrm{κ}$ and ${\mathrm{κ}}_c$ in ${\mathbb{R}}^n$ can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities $\mathrm{γ}$ and $\mathrm{Î\pm }$ (resp.). In this article we provide a dual characterization of ${\mathrm{κ}}_c$ in the spirit of the classical one for the capacity $\mathrm{Î\pm }$ by means of the Garabedian function. Using this new characterization, we show that $\mathrm{κ}(E)=\mathrm{κ}({∂}_{o}E)$ for any compact set $E⊂{\mathbb{R}}^n$, where ${∂}_{o}E$ is the outer boundary of $E$, and we solve an open problem posed by A. Volberg, which consists in estimating from below the Lipschitz harmonic capacity of a graph of a continuous function.

We obtain new Bass-Serre-type rigidity results for ${\mathrm{II}}_1$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any nonamenable factor arising as an amalgamated free product of von Neumann algebras $M_1{*}_B{M}_2$ over an abelian von Neumann algebra $B$ is prime, that is, cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\mathrm{II}}_1$ and in the type $\mathrm{III}$ cases, new examples of prime factors.

We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^{\infty }$-rings that is obtained by patching together homotopy zero sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection $A\cap B$ of submanifolds $A,B\subset X$ exists on the categorical level in our theory, and a cup product formula $[A]⌣[B]=[A\cap B]$ holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.

Algebraic curves have a discrete analog in finite graphs. Pursuing this analogy, we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are $2$-isomorphic. In particular, the strong Torelli theorem holds for $3$-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. By contrast, we prove that, in a suitably defined sense, the tropical Torelli map has degree one. Finally, we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.

We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of $3$-torsion elements in the class groups of quadratic fields. In addition, we prove analogous error terms for the density of discriminants of quartic fields and the mean number of $2$-torsion elements in the class groups of cubic fields. These results prove analytic continuation of the related Dirichlet series to the left of the line $R(s)=1$.