Splinter Group:
Geometry and Topology

Organizer: Javier Aramayona (NUI Galway)

All talks will be held in the Larmor Lecture Theatre.

Tuesday, April 7 2009

14:15 - 15:10

Peter Kropholler (University of Glasgow):
TBA

15:15 - 16:10

Ian Leary (Ohio):
Variations on a theme of Kan-Thurston

I shall state the Kan-Thurston theorem and some other related results, and explain how CAT(0) cubical complexes can be used to prove strong versions of these theorems.

16:30 - 17:00

Peter Giblin (University of Liverpool):
Vertices, inflexions and symmetry sets

This is joint work with Ricardo Uribe-Vargas and Andr\'{e} Diatta. Consider a smooth surface M in 3-space, and a point P of M. The tangent plane to M at P meets M in a singular curve which generically is an isolated point, two smooth transverse branches, an ordinary cusp or two smooth tangential branches (the latter occurs at a hyperbolic cusp of Gauss, or godron). Now move the tangent plane parallel to itself a small distance and consider the intersection. How many vertices (critical points of curvature) and inflexions (zeros of curvature) are there? And, somewhat harder, what is the 'symmetry set' of the resulting curve (that is, the locus of centres of circles bitangent to the curve)? I shall talk about some answers to some of these questions.

17:00 - 17:30

Thomas Cuschieri (University of Warwick):
Asymptotic Boundary Problems in Hyperbolic Space

The asymptotic Plateau problem for constant mean curvature (CMC) surfaces in hyperbolic 3-space involves finding a CMC surface whose ideal boundary is some prescribed Jordan curve on the sphere at infinity. The problem can be tackled using geometric measure theory, but one must then forfeit control over the topology of the resulting surface. In this talk I will describe the parametric approach, which yields a CMC disk as solution. I will then look at the issue of smooth dependence on the boundary data, and explain how one is lead naturally to seek a perturbation result for proper harmonic maps, a result which in turn hinges on a careful analysis of the Jacobi (second variation) operator for harmonic maps between hyperbolic spaces.

Wednesday, April 8 2009

14:15 - 15:05

Jim Howie (Heriot-Watt University):
Classifying spaces for residually free groups

I will describe some joint results with Martin Bridson, Chuck Miller and Hamish Short on the classification and homological finiteness properties of finitely presentable residually free groups. I will explain why such groups almost never admit compact classifying spaces, and discuss a conjectured criterion for the existence of a classifying space with compact k-skeleton for a given k.

15:05 - 16:00

Kurt Falk (NUI Maynooth):
Hyperbolic manifolds with dimension gap

Recurrent dynamics in geometrically finite hyperbolic manifolds is well understood by means of Patterson-Sullivan theory. For geometrically infinite manifolds the focus shifts to nonrecurrent dynamics, with mutually singular conformal measures associated to different infinite ends of a given manifold. Even more, nonrecurrent dynamics becomes the "thick part" of dynamics, not only in the sense of measure but also Hausdorff dimension, for manifolds with a dimension gap between recurrent and nonrecurrent dynamics. I will present some classical results alongside with newer research I was involved in.

16:15 - 16:40

Thomas Murphy (University College, Cork):
Einstein four manifolds of cohomogeneity one

The classification of compact Einstein four manifolds admitting an isometric action of cohomogeneity one has long been of interest to geometers. We present an overview and then discuss some recent progress on this question.

16:40 - 17:05

Mercedes Jordan-Santana (Dublin):
Invariants of Virtual Knots

A classical knot is an embedding of the circle in \Bbb R^3. For every diagram of a knot, we can obtain its Gauss code by assigning numbers to its crossings and listing them in an order obtained from the diagram. Then, for every diagram of a classical knot, we have a sequence of numbers although the reverse is not always true. Kauffman extended classical knots to virtual knots in order to have a solution for every sequence of numbers. In the same way as classical knots are defined by the 3 Reidemeister moves, there is also a generalization of the Reidemeister moves to virtual knots. I will consider examples of these knots and give the definition of a biquandle and show how it can be used as an invariant for virtual knots.
  

17:05 - 17:30

Nikos Georgiou (Institute of Technology Tralee):
The geometry of the space of oriented geodesics in hyperbolic 3-space

In this talk we introduce a Kahler structure on the space of oriented geodesics of the hyperbolic 3 space and we investigate its submanifolds.