Laminations: geometry, harmonic analysis and ergodic theory

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Proposal: DMS-9973086PI: Alberto CandelAbstract: This project proposes to study several problems in the area oflaminations/foliations. One of these problems is concerned with thegeometric structure of the leaves of a lamination. Some naturalquestions that the proposer intends to study are: what do the leavesof a lamination look like? how many different leaves(e.g. quasi-isometry classes) are there? what geometric invariants canbe (and cannot be) used to distinguish them? Still concerned with thegeometry of the leaves, there are questions about two dimensionallaminations and metric uniformization of their leaves. Especiallyinteresting are those which have a Euclidean conformal structure,which are often encountered in problems of low dimensional topologyand geometric group theory. Other part of this project falls in thearea of rigidity of virtual subgroups of semisimple groups andharmonic analysis on them. Research in this area has been concernedwith virtual subgroups which have an invariant measure. The proposerseeks to develop the study of harmonic measures, which are noninvariant in a reasonable way, and how they fit in the existing bodyof work on rigidity. Other part of the project concerns the study thatcertain geometric properties of the leaves of a foliation have on itstransverse or dynamical structure and on the topological structure ofthe ambient manifold.The research of the proposer is in the general area of qualitativestructure and behavior of dynamical systems. Dynamical systems areused to model processes in many areas, for example the evolution ofliving organisms and their morphology, the weather or other physicalor chemical processes. Other systems of interest come from processeswhich evolve from a finite amount of data according to some set ofrules, either specified before hand or of a random nature. This couldbe neural networks in the brain, systems of digital processors in acomputer, branching processes (as in nuclear fission, evolution andgrowth of cells, etcetera). Even when appearing under differentcircumstances, their dynamic behavior and structure follows similarlaws dictated by the individual elements and the nature of theirinteraction. The purpose of the researcher is to study and classify,from a geometric/qualitative viewpoint, invariants of these systemsrelating to their long time asymptotic behavior, their dependence onthe initial data, the generic properties of the possible states, andtheir global structure, all which can be used to understand themechanisms underlying such processes.