Grant: Discovery Grants Program - Individual

The concept of a "group" has become fundamental in modern mathematics and physics for the good reason that it captures in precise mathematical form the idea of symmetry. Each object, or instance of a mathematical concept, has its associated group of symmetries. Important among groups are the "free" groups, largest in the sense that they encode all groups in a standard, albeit subtle, way. My project concerns these, and allied, analogously "large", group constructions, such as the "amalgamated product", "square product", and a new product I call the "braid product" of two groups since it represents a generalization of the famous Artin braid groups. I plan to look at certain problems connected with these products. I also plan to continue work with my current Ph. D. Student on the "Andrews-Curtis conjecture", significant for the study of 3-dimensional surfaces, and with my collaborator Y. Medvedev on a well-known problem concerning "laws" in groups.