During the 2015-16 academic year, the School will have a special program on Geometric Structures on 3-manifolds, and Ian Agol of the University of California, Berkeley, will be the Distinguished Visiting Professor.
Thurston proposed the classification of geometric structures on n-manifolds. While the spectacular Geometrization Theorem classified the geometric structures on 3- manifolds with compact isotropy group, i.e. locally homogeneous Riemannian metrics, there is a cornucopia of other fascinating structures such as contact structures, foliations, conformally flat metrics and locally homogeneous (pseudo-) Riemannian metrics.
The goal of this program is to investigate these other geometric structures on 3-manifolds and to discover connections between them. Additionally, it is important to forge connections between geometric structures on 3-manifolds and other geometric constructs, such as gauge theory, PD(3) groups, minimal surfaces, cube complexes, geometric structures on bundles over 3-manifolds, and strengthened structures such as taut foliations, tight contact structures, pA flows, convex projective structures and quasi-geodesic foliations. Many of thse do not even have a conjectural classification (in terms of topological restrictions and moduli), and specific examples are still being constructed.
Two special program workshops will be held during term I. The first workshop, "Geometric structures on 3-manifolds", will take place during the week of October 5, 2015. The second workshop on "Flows, foliations and contact structures" will be during the week of December 7-11, 2015.
The goal of the October workshop will be to explore the classification of geometric structures on 3-manifolds, broadly interpreted. Constraints and consequences for the topology of manifolds with a given geometric structure, and connections between different geometric structures will be investigated.