In this period, ideas and techniques from Ergodic Theory have been percolating in many areas of mathematics and of applications. This project aims to expanding these interactions and to investigating the related problems in Ergodic Theory.
Of particular interest will be the horocycle flow on infinite type surfaces and the related questions in analytic number theory and on Teichmueller spaces, the noncommutative ergodic theorems in relation with geometric group theory and the ergodic theory of partially hyperbolic dynamical systems.
Most of the work of the PI has been devoted to understanding properties of mathematical entropy in geometry, dynamics, geometric group theory and probability. Mathematical entropy is a measure of complexity of infinite systems at a finite scale. This notion comes from information theory originally, but it has proven to be extraordinarily flexible. Often, and this is what the PI explores, precise comparaison of the entropy with other global quantites allows to recognize hidden symmetries. For instance, the PI (with Lin Shu, from Peking University) has characterized symmetric spaces (these are spaces with as many isometries as possible) among Riemannian manifolds by a geometric general condition (no focal points) and the equality of two entropies. The PI had before proven (with Xiaodong Wang, Michigan State University) a stronger characterization of hyperbolic spaces by the equality case in still another inequality between entropies. Such rigidity results are proven here for Riemannian manifolds, but there are analogous questions for groups or for foliations. Looking for other extremal situations, the PI also proved regularity results for the entropy of random walks on hyperbolic groups. The PI spreads this knowledge and his discoveries by delivering lectures at seminars and conferences, organizing conferences, giving mini-courses at Summer schools, advising students, answering questions and in a general way by sharing information through all communication means available.
