The investigation of algebra and number theory in the 19th and the 20th centuries raises some important historical issues. Although prominent algebraists and number theorists such as Dickson made extensive references to papers published in France, and despite the roles played by algebra and arithmetic in the development of the American mathematical community, our knowledge of these French works and authors is still very limited. In particular, we know almost nothing about the circulations of algebraic and arithmetic knowledge and practices between France and the United States at the beginning of the 20th century, or about the roles such circulations played in the development of international disciplines such as linear algebra in the 1930s. Here, our lack of knowledge is due in part to a combination of social and cultural factors within these contexts, a problem which raises difficult methodological issues. In order to tackle these issues, it is necessary to appeal to knowledge, sources and methods which cross both institutional (and furthermore national) and disciplinary frameworks. One way to approach this is through the development of significant collaborations between young scholars in Europe and America. Young researchers in the history of mathematics are often isolated in their institutions, and it is therefore another aim of this project to establish some modalities of collective researches that would endure over time. Instead of pursuing a global approach to the historical questions posed above, we have decided to focus on a few relevant contexts in order to develop some detailed and precise studies about the circulation of algebra and number theory. 1. The relationships between France and the USA. The first task of this project is devoted to the circulations of practices and knowledge in group theory and arithmetic between France and the United States. In so doing, it will be necessary to investigate sources which to date have not yet received adequate historical attention, including, among others the published and unpublished works of both famous authors (such as Dickson and Jordan) and authors (such as Séguier and Autonne) who are lesser-known today, despite being referred to extensively in the treaties of the 1930s that developed linear algebra as a mathematical discipline. We thus aim also to highlight some new sources and authors in order to analyze them and to understand how they formed a part of various networks and communities. 2. The situation in France. In analyzing this context we aim to identify the various algebraic and arithmetic written works in France as well as the networks in which these researches circulated. In so doing it is necessary to describe the collective phenomena of circulations in order to highlight individual achievements of mathematical creation. 3. Investigation of the identities taken on by algebraic and arithmetic practices. Between 1870 and 1945, algebra and arithmetic took on various and changing identities (in terms of their theories, practices etc.) depending on the places, communities and networks where they were studied. It is our aim to explore how certain algebraic practices circulated between and interacted with various disciplines, specifically quantum mechanics and operator theory, which in turn will shed light upon the evolutions of these disciplines as algebra progressively took on a fundamental role in the organization of mathematical knowledge. 4. Evolutions of mathematical disciplines and general issues on circulations. This portion of the project is intended to coordinate the findings of the three other case studies in order to develop some general results on the circulation of algebraic and arithmetic knowledge and practices (journals, translations, correspondences ...). These results will provide us with a new understanding of the structural role algebra came to play in the 1930s as well as the other images of this discipline that coexisted during that period.