Symmetry plays an important role in mathematics, physics, and other sciences. In mathematics, symmetry has many different but related manifestations. A finite group G of symmetries of a space X induces symmetries on algebraic invariants of X. In this way, transformation groups and representation theory become intimately related. In this project, the investigator will look systematically at the interaction between the geometry of the symmetry (e.g., fixed points, stability subgroups, etc.) and the algebraic invariants of such representations. In particular, symmetries of Moore spaces (the Steenrod Problem posed in early 1960's) and algebraic curves and surfaces arising in algebraic geometry and number theory (besides topology) provide concrete problems to which the following general theory will apply. A significant feature of this project is its interdisciplinary nature, which requires tools and ideas from several branches of mathematics. This provides a natural setting for interaction of these areas, including new applications of algebraic geometry to transformation groups and modular representation theory. Other applications include (algorithmic and inductive) construction of important classes of modular representations, number theoretic properties of algebraic function fields, and a deeper understanding of symmetries of algebraic surfaces versus 4- dimensional manifolds. A long term objective is solution of the long-standing Steenrod Problem and its application to the problem of existence of symmetry for manifolds.
