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Some Topological Methods in General Relativity and Multivariate Statistical Analysis
by
Steve Watson
York University
The infinitary methods of topology can be applied in a useful way to the natural and social sciences. In this lecture, we describe some essentially topological problems which arise in general relativity, statistical mechanics and multivariate statistical analysis of psychological data.
In general relativity, the problem of representing the Lorentz group of Minkowski space-time as the group of homeomorphisms of the space-time equipped with some reasonable topology remains unsolved despite the partial success of the (non-regular) path topology of Hawking, King and McCarthy. The problem of constructing a reasonable boundary for a space-time has also not been solved although the (possibly non-Hausdorff) b-boundary of Schmidt and the (possibly degenerate) causal boundary of Penrose, Kronheimer and Geroch are partial solutions.
The problem of treating error and noise in distance data and classifying and representing this data with Euclidean and tree models has applications in spin glass theory in statistical mechanics and the statistical analysis of dissimilarity data in psychology. The problem of obtaining exact solutions in two-dimensional scaling was solved by Menger in 1928 for the l2-norm (Euclidean-distance), by Malitz in 1992 for the l1-norm (taxi-cab metric), and in additive trees by Buneman in 1971 but the problems of obtaining exact solutions for representation in regions in the plane or in three-dimensional scaling with the l1-norm have not been solved. The problems of optimization of ultrametric models or additive tree models remain open under any reasonable criteria.
We give some purely topological versions of these basic scientific problems.
This work has been supported by the Natural Sciences and Engineering Research Council of Canada.
Date received: April 12, 1996
Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-92.