V has subgroups that are so close to being a BN-pair that the classical proof for simplicity of linear groups with irreducible Coxeter system goes through almost without change. It turns out that the subgroup F plays the role of the solvable Borel subgroup. [joint work with Jim Belk]

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure M on the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation identities for the Rogers dilogarithm function on the moduli space of a surface.

The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an open question; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n>=4. This differs from the behavior for n=2 (when the Dehn function is linear) and for n=3 (when it is exponential). In this talk, I will discuss some of the background of the problem and sketch a proof that the Dehn function of SL(n,Z) is at most quartic when n >= 5.

Picard modular surfaces are the non-compact arithmetic complex hyperbolic 2-orbifolds. I will prove that the two orbifolds studied by John Parker as candidates for orbifolds of smallest volume are indeed the unique arithmetic complex hyperbolic 2-orbifolds of minimal volume. Given time, I will also make some remarks on finding minimal volume manifolds.

Advanced techniques for understanding large scale scientific data are a crucial ingredient in modern science discovery. Developing such techniques involves a number of major challenges in management of massive data, and quantitative analysis of scientific features of unprecedented complexity. Addressing these challenges requires interdisciplinary research in diverse topics including the mathematical foundations of data representations, algorithmic design, and the integration with applications in physics, biology, or medicine. In this talk, I will present a set of case in the use of Morse theory for the representation and analysis of large-scale scientific data. Due to the combinatorial nature of the approach, we can implement the core constructs of Morse theory without the approximations and instabilities of classical numerical techniques. We use topological cancellations to build multi-scale representations that capture local and global trends present in the data. The inherent robustness of our combinatorial algorithms allows us to address the high complexity of the feature extraction problem for high-resolution scientific data.