When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph's boundary path groupoid.This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras.

We study strongly graded groupoids, which are topological groupoids G equipped with a continuous, surjective functor κ : G → Γ, to a discrete group Γ, such that κ −1 (γ)κ −1 (δ) = κ −1 (γδ), for all γ, δ ∈ Γ. We introduce the category of graded G-sheaves, and prove an analogue of Dade's Theorem: G is strongly graded if and only if every graded G-sheaf is induced by a Gε-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras.

BackgroundMany studies have shown that women who have experienced intimate partner violence (IPV) are at a greater risk of HIV, but the factors accounting for this association are unclear, and trials of interventions to reduce IPV have not consistently reduced HIV incidence.MethodsThis study uses an agent-based model, calibrated to South African data sources, to evaluate hypotheses about likely causal pathways linking IPV, HIV, and other confounding factors. Assumptions about associations between IPV and HIV risk behaviours were based on reviews of international literature.FindingsThere is an association between past IPV experience and HIV incidence even when no causal effects are assumed (IRR 1.28, 95% CI 1.23–1.34), because women with a propensity for multiple partners are more likely to have ever been in a relationship with a violent partner. If, in addition, men with a propensity for concurrent relationships are more likely to perpetrate IPV, the IRR increases to 1.42 (95% CI 1.36–1.48), consistent with empirical IRR estimates. Alternative scenarios in which experience of IPV is assumed to cause changes in women's sexual behaviour have little effect on the IRR. An intervention that reduces IPV by 50% could be expected to reduce HIV incidence by at most 1.3%.InterpretationMuch of the observed association between IPV and HIV is likely to be due to confounding behavioural factors. Although interventions to reduce IPV are important, these interventions alone are unlikely to have a substantial impact on HIV incidence.