Exotic differentiable structures and general relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (fake or exotic) differentiable structures on topologically simple manifolds such asS ⁷, ⁴ andS ³ X ¹. Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.     

On a new method for constructing good point sets on spheres

We use quadrature formulas with equal weights in order to constructN point sets on spheres ind-space (d 3) which are almost optimal with respect to a discrepancy concept, based on distance functions (potentials) and distance functionals (energies). By combining this approach with the probabilistic method, we obtain almost best possible approximations of balls by zonotopes, generated byN segments of equal length.Editors' note: We learned with sadness of Gerold Wagner's untimely death as a result of an avalanche in the Alps shortly after the submission of this paper. When one of the referees, Joram Lindenstrauss, suggested that Wagner's results might be extended to dimensions >6, we invited Professor Lindenstrauss to submit a paper containing that extension which we would publish alongside the Wagner paper. The result is the paper by Bourgain and Lindenstrauss that follows the present one.     

Order series of labelled posets

Order series of labelled posets are multi-analogues of the more familiar order polynomials; the corresponding multi-analogues of the related representation polynomials are calledE-series. These series can be used to describe the effect of composition of labelled posets on their order polynomials, whence their interest for us. We give a reciprocity theorem forE-series, and show that for an unlabelled poset the form of theE-series depends only upon the comparability graph of the poset. We also prove that theE-series of any labelled poset is a rational power series (in many indeterminates) and give an algorithm for computing it which runs in polynomial time when the poset is strictly labelled and of bounded width. Finally, we give an explicit product formula for theE-series of strictly labelled interval posets.This research was supported by the Natural Sciences and Engineering Research Council of Canada under operating grant #0105392.