Discrete transformation groups

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One of the unifying principles in geometry is that complex systems, such as configurations of planets and stars can often be understood by means of their symmetries. Familiar symmetries include the rotations or reflections of solids in space and the Lorentz transformations of space-time. Discrete invariants and groups of symmetry of continuous motions are studied in algebraic topology, while geometric topology is concerned with the properties of differential manifolds, or higher-dimensional surfaces. Geometry and topology is a flourishing subject for research, with broad connections to other areas of mathematics, science and engineering. Symmetries of manifolds are related to algebra and number theory through group theory, and to partial differential equations and analysis through differential forms. This proposal describes my recent work in three main areas (i) finite group actions on products of spheres, (ii) smooth and continuous group actions on 4-dimensional manifolds and their connections to gauge theory, and (ii) infinite discrete group actions on high-dimensional manifolds. The new projects include the development of a new coarse geometry for discrete group actions, a comparison of finite groups of differentiable transformations with those defined by algebraic equations on algebraic surfaces, and the study of dynamical and ergodic aspects of infinite groups of transformations on high-dimensional spheres. The goal in each case is to improve our understanding of the many roles of symmetry in mathematical and physical problems. This research proposal provides many high-level opportunities for training of prospective graduate students and postgraduate researchers, whose future work will have a broad impact on our society.