Connected Isotropy Groups in the Grove Symmetry Program

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The PI seeks to advance understanding of the interaction between the local, geometric properties and the global, topological properties of smooth objects called manifolds. This continues work that goes back over 150 years to the origins of Riemannian geometry, and it involves questions asked decades ago that still have no answers. Fortunately viewing some of these problems through the lens of symmetry in the last 25 years has led to major advances, the development of new tools, and the construction of new examples. The PI collaborates widely and plays his part in tying together the world’s people and their economies. With an eye to the future, the PI is also actively involved in the growth, education, and diversification of tomorrow’s body of U.S.-based, STEM-focused researchers. In the past ten years, the PI and his collaborators have proved striking new results in the Grove symmetry program. This program was developed in the 1990s to provide a foothold into the basic but difficult questions in Riemannian geometry around positive curvature. These new results have required new tools. For example, the PI and his collaborators have recently obtained surprising results for torus representations with connected isotropy groups. The proofs use only elementary representation theory. In particular, they do not involve curvature assumptions and might therefore have wider applications in other areas of geometry. Completing this work and further investigating its applications are top priorities for this project period.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.