In this project the PI proposes to study discrete geometry, curvature flows, and collapsing solutions to smooth geometric flows. The original motivation for this work is the landmark work on Ricci flow that began with R. Hamilton and includes the solution of the Poincare conjecture by G. Perelman. The PI proposes to work on both combinatorial curvature flows on piecewise linear manifolds and discrete approximations of Ricci flow and other smooth flows. In particular, the PI plans to study discrete flows numerically, to develop visualization techniques for abstract manifolds, to further develop the theory of discrete geometries in the spirit of differential geometry, and to prove convergence of these geometries and related geometric operators to the continuum. The PI also plans to study geometric flows on generalizations of Riemannian manifolds, such as Riemannian groupoids, in order to better understand those flows at singularities. The experimental part of this proposal will be run by a laboratory of undergraduates supervised by graduate students.
The recent solution of the Poincare conjecture by G. Perelman both stunned and invigorated the mathematics community. The PI proposes to study similar techniques involving geometric flows in two settings: (1) Discrete Geometries, which may be applied both to other types of geometric questions and to mathematical modelling in a variety of settings, including physics and computer graphics, and (2) Generalized Geometries, which may clarify the implications of Perelman's results and how it may be applied to both mathematical and physical applications. The hope is not only to solve geometric problems, but develop techniques applicable to other areas of science and engineering, both theoretically and computationally. In the process, the PI plans to rely on the laboratory science model to form a group of graduate and undergraduate students developing research tools and presentation tools. The PI hopes to use these tools to communicate the excitement of modern geometry to researchers, teachers, students, and the general public.
