In the moduli space $ \mathcal{M} $_g of genus-g Riemann surfaces, consider the locus $ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $ of Riemann surfaces whose Jacobians have real multiplication by the order $ \mathcal{O} $ in a totally real number field F of degree g. If g = 3, we compute the closure of $ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $ in the Deligne–Mumford compactification of $ \mathcal{M} $_g and the closure of the locus of eigenforms over $ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $. Boundary strata of $ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $ are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.

We apply this computation to give evidence for the conjecture that there are only finitely many algebraically primitive Teichmüller curves in $ \mathcal{M} $₃. In particular, we prove that there are only finitely many algebraically primitive Teichmüller curves generated by a 1-form having two zeros of order 3 and 1. We also present the results of a computer search for algebraically primitive Teichmüller curves generated by a 1-form having a single zero.

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.

This paper provides the details of the results announced in Christensen et al. Proc. Natl. Acad. Sci. USA 107 (2010), 587–591.

If X is a closed subset of the real line, denote by G_X the supremum of the size of the gap in the Fourier spectrum of a measure, taken over all non-trivial finite complex measures supported on X. In this paper we attempt to find G_X in terms of X.