Description:

This RIT will concentrate on the geometry and dynamical properties of moduli spaces of geometric structures, particularly on 2-dimensional manifolds. Our investigation focuses on the following general problems:

1. Describe the global topology (e.g. connected components, Betti numbers) of character varieties of surface groups. Interpret topological properties in terms of dynamical objects (flows, foliations, submanifolds, tensor fields) inside the moduli spaces.

2. Determine criteria for when a surface group representation is a discrete embedding, and more generally arises as the holonomy representation for a (possibly singular) uniformization of the surface.

3. Develop a calculus of these singularities to apply to produce invariants of the representation of a surface group. Harmonic map theory provides singular structures for a huge class of representations. An interesting invariant is the energy function on Teichmüller space associated to a representation of the fundamental group. Its qualitative properties mirror the dynamical complexity of the action of the mapping class group.

4. Relate the ergodic theory and topological dynamics of the mapping class group action to the topology and geometry of the moduli spaces. For example, regions where the mapping class group acts properly seem to correspond to nonsingular uniformizations (Teichmüller space, deformation space of convex real projective structures and Hitchin's generalization to split real forms). In these cases the energy function is a proper Morse function and the moduli space is contractible. Apparently trivial dynamics accompanies trivial topology.

Prerequisites:

The techniques will involve a combination of differential geometry, low-dimensional topology, complex analysis, commutative algebra, geometric group theory and ergodic theory. There are no course requirements, but proficiency in the material covered in the basic courses (Math 403, 410, 405, 414, 463 for undergraduates, 600, 630, 642, 660, 730, 744 for graduates) is expected. The main requirement is mathematical maturity.

Graduate Students:

This is an active and fertile area of current research. My home page lists titles for some of the theses I've directed in this area, as well as some of my own publications.

Undergraduate Students:

Visualization projects in the Experimental Geometry Lab use computer graphics to depict both structures and moduli spaces. Computer experience is desirable but not required. Previous projects have involved programming in C, Tcl, Java, Maple and Mathematica.

Mode of Operation:

In the biweekly meetings (10:00 Mondays in fall, time for spring semester yet to be determined), students will be expected to give presentations. Clear and informative documentation of the experimental aspects of the projects will be emphasized.