The main thrust of the proposed research is to introduce and develop new analytic techniques in group theory and the study of group actions. Major projects include: (a) studying quasi-isometries using a new notion of coarse differentiation (introduced in recent joint work with Eskin and Whyte) (b) developing harmonic map techniques in rigidity theory, particularly in the context of infinite dimensional target spaces and (c) developing a normal form theory for group actions based on new generalizations and variants of hard implicit function theorems.
All research lies in the broad interdisciplinary area of rigidity in dynamics, geometry and topology. The grant will also support summer schools devoted to new developments in rigidity theory. The field is broad and not entirely well defined, new developments frequently involve ideas from other areas of mathematics. These summer schools will provide students and young scientists the opportunity to learn about new developments quickly and to build the professional networks required to remain abreast of future new developments.
In the study of mathematical objects, a key role is often played by the symmetries of the object -- particularly when the object has many symmetries. The PI investigates ways of characterizing, describing and studying spaces with many symmetries in various dynamical, geometric and topological settings. These questions are interdisciplinary in nature and often require learning, adapting and applying ideas and techniques from many areas of mathematics. The PI's work has connections with diverse areas of mathematics: from celestial mechanics (KAM theory and stability of the solar system) to theoretical computer science (expander graphs, Kazhdan's property (T) and coarse embedding problems). Many proposed projects involve applying analytic ideas and techniques to problems traditionally studied by dynamical, geometric or topological methods.
The main focus of this project was the study of rigidity theory in geometry and dynamics. One can ask when an object is recognizedby relatively little information about the object. And one canask which particular class of objects is easily recognized fromlittle information. Rigidity theory is primarily interested in thelater form of question. A principal question of study is whena highly symmetric geometry (a homogeneous space or finitely generated group) is easily recognized when seen from far awayor with blurry vision (is quasi-isometrically rigid). Thisprogram of study was introduced by Gromov in 1983 and hasreceived attention from many researchers. In work with Eskin and Whyte, the PI introduced a new approachto this kind of question using a notion of coarse differentiation.This is a new notion that has since had other applications inboth geometry and theoretical computer science. It involvesa type of multi-scale analysis that is not unusual in mathematics,but it involves applying those notions in contexts that are quite novel. In particular, the usual notion of differentiationis applied only to maps that are very regular at small scales and ournotion applies well to maps that are not even well defined at smallscales. The main goal is to find any structure of the map that sends``straight lines" or geodesics close to geodesics. Another direction of research studied under this grant is rigidity of largegroup actions on compact manifolds. In this setting one asks when a relatively small object with many symmetries is completely characterizedby those symmetries. In recent work with Kalinin and Spatzier, thePI introduced new analytic techniques that allowed for several majorbreakthrough. This includes work by Rodriguez-Hertz and Wang resolvinga longstanding conjecture of Katok and Spatizer. The new technique involvesanalyzing regularity of functions using so called "wave front sets". While standard in some areas of mathematics, this is novel in this one, and themain insight involves connecting this notion to a certain kind of randomness in dynamical systems, the so-called exponential decayof matrix coefficients. The combination of these two tools should havemany further applications.
