My current research interests are centered on different aspects of the geometry of surfaces and hyperbolic geometry.
My personal page can be found at www.multijimbo.com
- Hyperbolic space and its isometries
- Kleinian groups (discrete groups of isometries of hyperbolic 3-space)
- Signatures of actions of automorphism groups on Riemann surfaces
The primary focus of Jim's research concerns hyperbolic space and its isometries. Specifically, Jim has spent much of his career investigating Kleinian groups, which are discrete groups of isometries of hyperbolic 3-space.
One specific question Jim has engaged with somewhat extensively is the limit set intersection theorem for Kleinian groups, which are the theorems that describe the difference between Λ(θ ∩ Φ) and Λ(θ) ∩ Λ(Φ) for subgroups θ and Φ of a Kleinian group Γ, subject to different hypotheses on Γ, θ and Φ. In dimension 3, this question is, Jim feels, essentially answered. There is still room for exploration for this question for Kleinian groups acting on higher dimensional hyperbolic space.
Another area of more recent focus is joint work with Dr Aaron Wootton (University of Portland), investigating signatures of actions of automorphism groups on Riemann surfaces (see below).
At present, Jim has several projects underway:
1. Schottky groups are a particular class of Kleinian groups acting on hyperbolic 3-space, which are algebraically straightforward (as they are free groups) and dynamically interesting. Schottky groups came in two flavors, classical (which can be defined in terms of pairings of circles) and non-classical (which cannot).
In this project, Jim investigates the extent to which subgroups of classical Schottky groups are or are not themselves classical, and the extent to which subgroups of non-classical Schottky groups are and are not themselves non-classical.
2. In joint work with Dr Aaron Wootton (University of Portland), we continue our investigation of aspects of signatures of actions of automorphism groups acting on closed Riemann surfaces, using the framework of skeletal signatures we introduced in our earlier papers.
3. A question that has captured Jim's imagination is the extent to which classical geometric theorems from Euclidean planar geometry hold in the hyperbolic plane.
Over the years, Jim has taught a variety of modules across Pure Mathematics. In particular, Jim developed a module on Hyperbolic Geometry, which led to the Hyperbolic Geometry textbook he wrote as part of Springer Verlag's SUMS series.
Currently, Jim is teaching MATH3033 Graph Theory, which introduces students to the basic definitions, concepts and theorems concerning graphs.
Jim graduated from the University of Georgia with a BA in Mathematics in 1986 and then undertook his PhD in Mathematics at the Stony Brook University, completing in 1991 under the supervision of Prof Bernie Maskit. You can find more information about Jim's mathematical family via the Mathematics Geneology Project.
After a postdoctoral year at MSRI in 1991-1992 and a G C Evans Instructorship in the Mathematics Department at Rice University from 1992 - 1995, Jim joined the School of Mathematical Sciences at the University of Southampton in 1995.