Geometric group theory, analysis on metric spaces, and geometric flows

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AbstractAward: DMS-0505610Principal Investigator: Bruce A. KleinerThe first part of the project concerns two geometric evolution equations:mean curvature flow and Hamilton's Ricci flow. The main objectiveis to understand the structure of the singular set and the geometry ofthe solutions near the singular set.The second part of the research program is motivated by geometricgroup theory and rigidity questions, specifically the asymptoticstructure of negatively curved (or Gromov hyperbolic) spaces.This leads to an investigationof spaces which have a self-similar character using analytictools that have been developed in the last few years. Herethe goal is to quasisymmetrically deform the space into an optimal form, if possible,in order to reveal hidden symmetries; show thatno hidden symmetries exist; or show that the original spacewas really a deformation of something familiar. This is directlyrelated to group theoretic rigidity problems.The project aims to study two nonlinear analogs of the heat equation: theevolution of surfaces by their mean curvature, and Hamilton'sRicci flow. Mean curvature flow is a standard (idealized) model formany physical process which involve an evolving surface, or interface,between two regions in space. The Ricci flow -- an equationgoverning a curved space geometry which evolves with time -- has made headlines lately due to itsprominent role in the spectacular work of Perelman on the 100 year old Poincareconjecture. Much is known about the solutions of these two equations,but many basic open questions remain, especially those tied with theformation 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 investigationof spaces which have a self-similar, or fractal character, using analytictools that have been developed in the last few years. Herethe goal is to deform the space into an optimal form, if possible,in order to reveal hidden symmetries, show thatno hidden symmetries exist, or show that the original spacewas really a deformation of something familiar. This is very usefulfor understanding the asymptotic shape of infinite groups of symmetries.