Principal Investigator: Bruce A. Kleiner
The first part of the project concerns two geometric evolution equations: mean curvature flow and Hamilton's Ricci flow. The main objective is to understand the structure of the singular set and the geometry of the solutions near the singular set. The second part of the research program is motivated by geometric group theory and rigidity questions, specifically the asymptotic structure of negatively curved (or Gromov hyperbolic) spaces. This leads to an investigation of spaces which have a self-similar character using analytic tools that have been developed in the last few years. Here the goal is to quasisymmetrically deform the space into an optimal form, if possible, in order to reveal hidden symmetries; show that no hidden symmetries exist; or show that the original space was really a deformation of something familiar. This is directly related to group theoretic rigidity problems.
The project aims to study two nonlinear analogs of the heat equation: the evolution of surfaces by their mean curvature, and Hamilton's Ricci flow. Mean curvature flow is a standard (idealized) model for many physical process which involve an evolving surface, or interface, between two regions in space. The Ricci flow -- an equation governing a curved space geometry which evolves with time -- has made headlines lately due to its prominent role in the spectacular work of Perelman on the 100 year old Poincare conjecture. Much is known about the solutions of these two equations, but many basic open questions remain, especially those tied with the formation of singularities, which play a central role in Perelman's work. The project will address some of these questions. Another component of the research program is an investigation of spaces which have a self-similar, or fractal character, using analytic tools that have been developed in the last few years. Here the goal is to deform the space into an optimal form, if possible, in order to reveal hidden symmetries, show that no hidden symmetries exist, or show that the original space was really a deformation of something familiar. This is very useful for understanding the asymptotic shape of infinite groups of symmetries.
