| Friday, May 8 | Saturday, May 9 | |
|---|---|---|
| 9:30am-10:30am | Michel Vandyck | François Labourie |
| 10:30am-11:30am | Stefan Bechtluft-Sachs | David Wraith |
| 11:30am-12:00pm | Coffee | Coffee |
| 12:00pm-1:00pm | Jesse Ratzkin | Andrei Mustaţă |
| 1:00pm-2:30pm | Lunch | Lunch |
| 2:30pm-3:30pm | Stephen Buckley | Tommy Murphy |
| 3:30pm-4:00pm | Coffee | |
| 4:00pm-5:00pm | Brendan Guilfoyle |
A number of variational problems for smooth maps of Riemannian manifolds only have solutions defined outside a low dimensional subset of the domain. One of the most prominent cases studied in solid matter physics is that of nematic liquids (liquid crystals). These are modelled by a manifold with boundary and a map to a projective space or bundle encoding the direction of the axis of the particles. If this map, i.e. the orientation of the particles, is prescribed on the boundary then the particles try to extend it over the whole manifold as smoothly as possible. Already for topological reasons this extension generally only exists up to a subcomplex of the manifold, the "defect" which is itself a closed submanifold in many cases.
In this talk we present some results on the homotopy theory of such defective maps. Sets of concordance classes and cobordism groups of defective maps are computed using classifying spaces. We also construct charactistic numbers to determine cobordism classes.
This is joint work with Simon Kokkendorff.
Nonpositive curvature conditions and notions of boundary at infinity are quite useful in various aspects of analysis on metric spaces. We discuss two notions of boundary at infinity in a metric space context: the ideal boundary and the conformal boundary. Under reasonable conditions, these concepts can be identified for Gromov hyperbolic spaces. We discuss the relationship between these two boundaries at infinity for general spaces. Although they can be very different from each other in CAT(0) spaces (as easy examples show), they are both simply connected for Hadamard manifolds of dimension at least 3. We characterize the conformal boundary of a warped product, thus providing some examples of CAT(0) spaces where the ideal and conformal boundaries differ in interesting ways.
In this talk I will present some recent work (in collaboration with Wilhelm Klingenberg) on mean curvature flow of n-dimensional spacelike submanifolds of a manifold endowed with a metric of signature (n,m). We establish long-time existence of the flow subject to mild curvature assumptions by proving a priori gradient estimates.
Given a representation of a surface group in SL(n,R), a classical result states that, under a mild hypothesis, there exists an unique equivariant harmonic mapping from the universal cover of the surface to the associated symmetric space. Therefore each representation of a surface group gives rise to a functional, called energy, on Teichmuller space which is the value of the energy of the corresponding harmonic map. For some representations, that we will call "displacing", we prove that this energy is proper, hence the existence of equivariant minimal surfaces. We shall discuss examples of these displacing representations.
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.
Birational geometry is the study of the various compactifications of a (non-compact) algebraic variety. In the category of smooth varieties, any two such compactifications are related by a sequence of elementary transformations called blow-ups. Thus, knowing how cohomological invariants like, e.g. the Chern polynomials, behave under blow-ups allows one to compare these invariants for two compactifications of the same variety.
From the point of view of cohomological invariants, a natural extension of the category of smooth varieties is that of smooth orbifolds (which can be viewed locally as quotients of smooth varieties by finite groups). I will explain why the natural analogs of blow-ups in this context are the weighted blow-ups along local embeddings. We compute cohomological invariants of such objects by extending the category of smooth orbifolds such that it will also include non-separated objects. This talk is based on joint work with Anca Mustata.
I will discuss some lower bounds for the first Dirichlet eigenvalue of the Laplacian for bounded domains in cones. These bounds arise from weighted isoperimetric inequalities. Time allowing, I will also discuss applications.
D-differentiation is a type of derivation of the algebra of tensor fields defined on a manifold, which contains (among others) Lie differentiation and covariant differentiation as special cases. It will be introduced in an elementary fashion and illustrated by examples taken from Physics. (Reference: "Topics in Differential Geometry; a new approach using D-differentiation", D. Hurley and M. Vandyck, Springer-Praxis 2002, ISBN 1-85233-491-6)
Existence questions for positive scalar, Ricci and sectional curvature metrics have been studied extensively. Much less is known, however, about the space of such metrics. In this talk we will focus on certain exotic spheres which are known to admit Ricci positive metrics. We will discuss the topology of the moduli space of Ricci positive metrics on these objects.
The Irish Geometry Conference 2009 is funded by the University College Cork School of Mathematical Sciences, the Irish Mathematical Society and the University College Cork Physics Department.
Visa invitation letters will not be provided.
| Telephone | 353 21 490 2434 |
| Fax | 353 21 427 0813 |
| Secretary | 353 21 490 2542 |
