The Complementary Program has a limited number of memberships that are open to mathematicians whose interests are not closely related to the core programs; special consideration is given to mathematicians who are partners of an invited member of a core program.

<p>This figure depicts dynamics of flows on convex cocompact hyperbolic 3-manifolds; where the girl is a traveller along a horocycle.</p>

This research program will bring together two intellectual communities that have made significant advances in the study of discrete subgroups of higher rank semisimple Lie groups: the homogeneous dynamics community and the community studying geometric structures and Anosov groups.

The stable and unstable foliations near a singular orbit of a pseudo- Anosov flow in 3 dimensions. Courtesy Michael Landry.

Low dimensional topology is a meeting place for many objects and ideas from diverse areas of mathematics, including foliation theory, geometry, and smooth and conformal dynamics.   For instance, many foliations on 3-manifolds admit transverse flows, connecting (local) leafwise homeomorphisms to flow dynamics and the mapping class groups of the leaves.  Leafwise conformal or hyperbolic structures can be approached through Teichmüller theory, and connect again to one-dimensional dynamics through "universal circles" organizing compactifications of all the leaves or of the flow space.  Many of these ideas originate in work of Thurston but in recent years have diverged and are ripe for reconnection.  

The program will bring together experts in all these fields together with younger researchers, who together can form new connections and open new areas for exploration.

Foliations around a pseudo-Anosov singularity (Image credit: Chi Cheuk Tsang)

This workshop will bring together ideas from diverse areas of mathematics that meet in the setting of geometry and topology in low dimensions.  This includes the study of flows, foliations, and fibrations of three-manifolds and the related study of geometry (e.g. hyperbolic or conformal structures) of the manifolds and of the leaves or fibers, and their mapping class groups.  This is a rich and interconnected area and many adjacent topics will also be featured.