A group of symmetries of a space X induces symmetries on certain algebraic invariants of X (so-called homology representations). In this way, transformation group theory (the study of the symmetries of continuous objects, such as spaces) and representation theory (the study of symmetries of discrete or algebraic objects) become intimately related. The goal of this project is to investigate systematically the interactions between the geometry of symmetries (such as fixed points) and their algebraic manifestations. In particular, the problem of existence of symmetries of Moore spaces with prescribed algebraic representations (the Steenrod Problem, posed in the early 1960's), algebraic surfaces, and certain 4-dimensional spaces all provide concrete problems to which a general theory should apply. Techniques from topology, algebraic geometry, and representation theory should interact fruitfully in this endeavor. Symmetry plays an absolutely fundamental role in science (in fact, in art and music as well), and we attempt to understand this role through qualitative and certain quantitative measurements and comparisons. In mathematics, symmetry plays a fundamental role also, and it has many different but related manifestations. To have a precise and logical understanding of symmetries of a mathematical structure, such as a space, we need to discover and quantify its so-called invariants and other important features. A wide variety of sophisticated tools of this trade, chiefly algebraic ones, have been developed over the years in different research projects, and this investigator is adept at and plans to draw upon a surprising number of them in his current work.
