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Organizers |
Calabi-Yau equations and 4 dimensional lattice Green functions
by
Tony Guttmann
Department of Mathematics and Statistics, University of Melbourne
Lattice Green functions in two dimensions are straightforward, and have now largely been solved for most three-dimensional lattices. In four dimensions there has been no complete solution. However recently a large family of 4th order ordinary differential equations, possessing maximal unipotent monodromy and satisfying the Calabi-Yau condition have been solved. We show that this enables us to obtain 4-dimensional lattice Green functions for the first time.
Date received: October 31, 2008
Harmonic deformations of hyperbolic 3-manifolds
by
Craig Hodgson
University of Melbourne
Coauthors: Steve Kerckhoff (Stanford)
This talk will give an introduction to our work on harmonic deformations of hyperbolic 3-manifolds, and describe some topological applications. In particular, this work gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem, including precise estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling.
Paper reference: math.GT/0301226
Date received: October 30, 2008
The p-HH-norm on the Cartesian product of n copies of a normed space
by
Eder Kikianty
Victoria University, Australia
Coauthors: Gord Sinnamon (University of Western Ontario, London, Canada)
The n-fold Cartesian product Xn of a normed space X is again a normed space when it is equipped with any one of the well-known p-norms, p ∈ [1, ∞]. These norms are equivalent but are not equal for different p. In 2008, Kikianty and Dragomir introduced the p-HH-norms on the vector space X2 of pairs of elements from X. These norms are equivalent to the p-norms but, unlike the p-norms, they do not depend only on the norms of the two elements in the pair, but also reflect the relative position of the two elements within the original space X. In this talk, I will discuss recent work in which the p-HH-norms are defined on the space Xn for n > 2.
As in the case of X2, the p-HH-norms on Xn depend on the relative positions in X of the elements in the n-tuple. They preserve the completeness of the space X. Also, they preserve geometric properties of the space X such as completeness, smoothness, Fréchet smoothness, strict convexity, uniform convexity, when 1 < p < ∞; and reflexivity, when 1 ≤ p < ∞. When the underlying space is the field of real numbers, the p-HH-norm on Rn is the hypergeometric mean of a positive n-tuple. Unlike the case of X2, the embeddings that establish the equivalence of the p-HH-norms to the p-norms in Xn have norms that depend on the space X, specifically, on the convexity of the unit ball in X.
Although the p-norms are all equivalent on the finite product Xn, they are all inequivalent as n goes to infinity and, consequently, they each determine a different normed subspace of X∞. When X=R these are the familiar lp sequence spaces. Similarly, the p-HH-norms give rise to new normed spaces of sequences in X and, when X=R, to new sequence spaces of real numbers. Comparing these new sequence spaces with those arising from the p-norms is an important area for future work.
Date received: September 10, 2008
Regularization of some equivariant Euler classes
by
Rongmin Lu
University of Adelaide
The theory of zeta-function regularization has grown from its appearance in the Ray-Singer definition of analytic torsion into a useful tool in mathematical physics. We propose a variant of zeta-function regularization - W-regularization - and apply this to some S1- and S1×S1-equivariant Euler classes, which are defined for certain infinite-dimensional vector bundles using an approximation technique. We find that we obtain new multiplicative genera and recover some familiar ones.
Date received: October 31, 2008
Minimal triangulations of 3-manifolds
by
J.H. Rubinstein
Department of Mathematics and Statistics, University of Melbourne
Coauthors: S. Tillmann, (Melbourne) and W. Jaco (Oklahoma State)
Using some new techniques involving labelling edges and tetrahedra, via Z2 torsion elements in homology, we have been able to classify the minimal triangulations of some infinite classes of spherical 3-dimensional manifolds. We are currently working on extending this to other types of geometric structures.
Paper reference: arXiv:0805.2425
Date received: October 29, 2008
Analytic torsion for twisted de Rham complexes
by
Mathai Varghese
School of Mathematical Sciences, University of Adelaide
Coauthors: Siye Wu (U. Colorado, Boulder)
We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by Ray-Singer, as it uses pseudodifferential operators. We show that this analytic torsion is independent of the choice of metrics on X and E, establish some basic functorial properties, and compute it in many examples. We also establish the relationship of an invariant version of analytic torsion for T-dual circle bundles with closed 3-form flux. This is joint work with Siye Wu.
Paper reference: arXiv:0810.4204
Date received: October 29, 2008
String Structures and Characteristic Classes for Loop Group Bundles
by
Raymond Vozzo
University of Adelaide
The string class of a loop group bundle P is the obstruction to lifting the structure group to the central extension of the loop group. The string class is related to the first Pontrjagyn class of a certain G-bundle associated to P. In this talk we will review the known results regarding this class and develop a notion of higher string classes for loop group bundles, which are associated to characteristic classes of certain G-bundles.
Date received: October 27, 2008
A new approach to hyperbolic geometry
by
Norman Wildberger
University of New South Wales
Hyperbolic geometry can be introduced in a simpler and more general way by using the framework of rational trigonometry and universal geometry within the Cayley Beltrami Klein model.
This way the subject extends to DeSitter space, incorporates a fundamental duality between points and lines, emphasises the importance of the null cone, and works over a general field. Many theorems of Euclidean geomety now can be given their proper Hyperbolic formulation, and the connections with Special Relativity become much clearer.
This talk will have lots of pictures.
Date received: October 31, 2008