Symmetry and Self-Similar Structures in Geometry and Topology

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Symmetry is a pervasive concept in mathematics, giving rise to beautiful and interesting geometric objects, as well as providing an essential tool in geometry and topology. Classically, research has focused on globally defined symmetries, but recently it has been realized that symmetries of local structures offer vastly more examples and connections to other areas of mathematics. This project will study local symmetry, its applications to the study of self-similar geometric objects and related algebraic problems, and recently discovered geometric constructions of efficient networks (so-called expander graphs).The relationship between the geometry of an object and the algebraic structure of its group of automorphisms has been intensely studied in the past, but in the last decade, progress has been made on a much richer theory of symmetries that only appear after passing to a cover, i.e. are "hidden'" The project aims to study this notion in several contexts, such as Riemannian geometry (especially nonpositive curvature) and geometric structures. Further, rich phenomena are expected when the symmetry arises from self-similarity, i.e. in the case of self-covering manifolds, particularly in situations of geometric interest such as Kaehler or affine geometry. Finally, the project aims to study coarse geometric objects associated to group actions (so-called warped cones), especially in the case of actions with spectral gap, where these yield expander graphs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.