Many optimization problems and engineering problems connected with linear systems lead to matrix inequalities. Matrix inequalities are constraints in which a polynomial or a matrix of polynomials with matrix variables is required to take a positive semidefinite value. Many of these problems have the property that they are "dimension free" and, in this case, the form of the polynomials remains the same for matrices of all size. In other words, we have noncommutative polynomials. One very much desires these polynomials to be "convex" in the unknown matrix variables, since if they are, then numerical calculations are reliable and local optima are global optima. In this dissertation, we classify all changes of variables (not containing transposes) from noncommutative non-convex polynomials to noncommutative convex polynomials. This introduces notions of noncommutative complex Hessians and plurisubharmonicity, classical notions from several complex variables. In addition, we present a theory of noncommutative integration and we prove a "local implies global" result in that we show noncommutative plurisubharmonicity on a noncommutative open set implies noncommutative plurisubharmonicity everywhere

Global health's goal to address health issues across great sociocultural and socioeconomic gradients worldwide requires a sophisticated approach to the social root causes of disease and the social context of interventions. This is especially true today as the focus of global health work is actively broadened from acute to chronic and from infectious to non-communicable diseases. To respond to these complex biosocial problems, we propose the recent expansion of interest in the field of global health should look to the older field of social medicine, a shared domain of social and medical sciences that offers critical analytic and methodological tools to elucidate who gets sick, why and what we can do about it. Social medicine is a rich and relatively untapped resource for understanding the hybrid biological and social basis of global health problems. Global health can learn much from social medicine to help practitioners understand the social behaviour, social structure, social networks, cultural difference and social context of ethical action central to the success or failure of global health's important agendas. This understanding - of global health as global social medicine - can coalesce global health's unclear identity into a coherent framework effective for addressing the world's most pressing health issues.