Aravind Asok, USC

Title: Constructing projective modules

Abstract:  An $n \times n$-matrix $P$ is called a projection matrix if $P$ is an idempotent, i.e., $P^2 = P$; in this context the notion seems anodyne.  If our matrices have coefficients in a more complicated algebraic structure, say a commutative ring $R$, then the image of a projection matrix is called a projective module.  I would like to explain the importance of social factors in the emergence of projective modules as an object of mainstream mathematical focus in the mid to late 1950s.  In particular, paraphrasing historian of science Massimo Mazzotti, I would like to explain the institutionalization of the notion of projective module by structuring its insertion into a ``network of concepts and practices supported by the collective interests of a social group''.  My main aim is to argue that ``mainstream focus'' in mathematics is socially directed.