Geometry and Topology of Manifolds

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The overall objective of my research program is the advancement of knowledge about the geometry and topology of manifolds. 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 active connections to other areas of mathematics, medicine, 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 recent work towards my long-term goals in four main areas (i) smooth and continuous group actions in dimensions 3 and 4, and their connections to gauge theory, (ii) classification of 4-manifolds with right angled Artin fundamental groups, (iii) finite group actions on products of spheres, and (iv) symmetries of smooth and topological Kervaire manifolds. Over the next five years I plan to open up several new directions, including the study of the space of homotopy self-equivalences of manifolds, the study of group actions in surface bundles over surfaces, and the development of local-global methods in surgery theory for infinite discrete groups. These lines of inquiry address basic problems, with a high potential for significant impact if the goals are achieved. This proposal offers high-level interdisciplinary opportunities for prospective graduate students and postgraduates, whose future work will shape our society. Geometric topology has applications in biology, through the knotting and linking of replicating DNA strands. Algebraic topology provides effective methods for analyzing large data sets, through "persistent homology". A geometrical perspective is essential: mathematical models in most realistic situations are multi-dimensional and involve spatial as well as temporal constraints (e.g. the recent "topology-preserving" neural net models for forest fire prediction and control). Fundamental research at Canadian universities is critical to providing the next generation of mathematicians and scientists. Our training activities, based on principles of equity and inclusion, will help to remove barriers to success and promote a more diverse workforce in a knowledge-based economy.