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Research Themes
Publications - Curriculum Vitae

Mathematics offers endless puzzles, challenges and new ideas to learn. Almost all of my research has used in some way techniques or ideas from the theory of foliations. A foliation is like an onion: you just peel space back layer by layer to see what's there.

My Curriculum Vitae gives a semi-complete list of my talks, which indicates the various areas of interest for my research.

Ergodic theory of foliations

The study of the dynamics of group actions and foliations presents many new ideas and difficulties. Traditional dynamical systems techniques are based on a singly generated system, and the most sophisticated methods require the existence of invariant measures. For foliations, the dynamics are defined by the local actions of the holonomy transformations, which provides a more complex system, and often precludes the existence of invariant measures. This is a topic rich in ideas and problems, where every insight opens new avenues of exploration.

Rigidity of group actions

Rigidity of group actions is a fundamental problem, proved using techniques from measurable dynamics, smooth hyperbolic dynamics, coarse geometry, and cohomology theory of representations.

Analysis on foliations

The leaves of a foliation are naturally endowed with complete Riemannian metrics and geometrically defined elliptic operators. The spectral theory of these operators, and the study of how the spectral theory is influenced bythe dynamics of the foliation, is a subject rich in results and new ideas. A special case of this is the study of the index theory for geometric first-order operators on leaves of a foliation, where the geometry at infinity of the leaves plays a central role. Foliation techniques yield simpler approaches to the Novikov Conjecture for compact manifolds and foliations. Index Theory for transverse operators ties together analytic invariants (eta invariants) of foliations and group actions with topological data. Some of the open questions concern.

Vanishing theorems for the secondary classes

The secondary classes of foliations combined analytic estimates with ideas of smooth dynamical systems to understand how the secondary characteristic classes of foliations are related to dynamical properties of its leaves. These classes remain ellusive in their geometric interpretation.

One of the great mysteries of the secondary classes is their role in the analysis of foliations. My work with Katok suggests some ties, and the relations between spectral flow and secondary invariants for foliations - as well as the very difficult results of Connes and Moscovici on the "Tranverse Fundamental Class" - hints at a much deeper theory waiting to be discovered.

Topology of foliations

Rational homotopy methods are very useful studying the homotopy types of the classfying spaces for foliations. Work on this area started with my thesis, and has continued since.

The leaves of a foliation form a veritable zoo of examples of open complete manifolds. Looking for themes and patterns by which to classify these leaves ups to coarse isometry, or by other topological properties is of great interest. The work with Oliver Attie on coarse types of manifolds which cannot be leaves was a notable step, though barely scratched the surface. More recent work with Hellen Colman on the category of foliations combines leaf topology with homotopy invariants.

A longer-term project is to give a unified exposition of the properties of the topology of foliation classifying spaces. There are now 25 years of varied studies of the topology of these spaces: work by Haefliger, Mather, Thurston, McDuff, Tsuboi, myself and others are separate islands in a larger sea of phenomenon associated with these universal spaces.

Updated April 27, 2017