This work applies the ideas of Alekseev and Meinrenken's noncommutative Chern-Weil theory to describe a completely combinatorial and constructive proof of the wheeling theorem. In this theory, the crux of the proof is, essentially, the familiar demonstration that a characteristic class does not depend on the choice of connection made to construct it. To a large extent, this work may be viewed as an exposition of the details of some of Alekseev and Meinrenken's theory written for Kontsevich integral specialists. Our goal was a presentation with full combinatorial detail in the setting of Jacobi diagrams. To achieve this goal, certain key algebraic steps required replacement with substantially different combinatorial arguments.

In the present paper we extend A. Weil's proof of the Siegel-Weil theorem to symplectic, nilpotent $t$-modules (snt-modules), $t$ being a nilpotent endomorphism of a finite-dimensional, symplectic, vector space satisfying a certain consistency condition with respect to the symplectic structure. This extension is key for our proof of the Siegel-Weil theorem for loop groups in the sequel to this paper.

Consider a free ergodic measure-preserving profinite action $\Gamma \curvearrowright X$ (i.e., an inverse limit of actions $\Gamma \curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally, of a group $\Gamma$ which admits an infinite normal subgroup ${\Gamma }_0$ such that the inclusion ${\Gamma }_0\subset \Gamma$ has relative property (T) and $\Gamma /{\Gamma }_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\Gamma \times X\to \Lambda$ is a measurable cocycle with values in a countable group $\Lambda$, then $w$ is cohomologous to a cocycle $w\prime$ which factors through the map $\Gamma \times X\to \Gamma \times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $\Gamma \curvearrowright X$ with any free ergodic measure-preserving action $\Lambda \curvearrowright Y$ comes from a (virtual) conjugacy of actions.

We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds and involve a proper notion of uniqueness for such solutions.

In our previous paper we studied some questions related to the $C^1$ and Lipschitz harmonic capacities. A serious error was found in the arguments. In this note we explain how this error, which we have not been able to fix, affects the results claimed in that paper.