SGER: The Algebraic Topology of Random Fields and its Applications

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The core of this project is the setting up of a research network, initially between four researchers, in three institutions, supporting interchange between them (Robert Adler, Technion, Jonathan Taylor, Stanford, Shmuel Weinberger and Keith Worsley, Chicago) and, more importantly, their students. They will study problems related to the geometric characterization of random structures, such as topological invariants of excursion (nodal) sets of random fields and learning the homotopy and homeomorphism types of manifolds in the presence of noise and developing techniques for quantifying the statistical accuracy of the resulting estimates. Whereas previous work on random field geometry has concentrated on characteristics coming from differential topology, such as Lipshitz-Killing curvatures (which include the Euler characteristic) this project will concentrate on characteristics coming from algebraic topology, involving homology-invariant concepts such as the number of connected components, Betti numbers, and other related invariants. Furthermore, while previous work in these areas was in the setting of Riemannian manifolds, it is planned to extend this to allowing non-Riemannian metrics even on smooth manifolds. In manifold learning this is important due to the fact that in some applications natural measures of closeness underlying the context of a data set can well be non-Riemannian. In the context of random fields, it may open new doors in the study of purely non-Gaussian fields, such as stable fields, where the natural geometry on the parameter manifold induced by the field is non-Riemannian.The motivation for this project, which links algebraic topology, probability, and pure and applied statistics comes from both its intrinsic mathematical interest and from a wealth of applications, from areas as broadly spaced as the statistical analysis of fMRI brain images, high dimensional data analysis in computer science, and cosmological projects such as the COBE and WMAP experiments and the Sloan digital sky survey. The four PIs come from a variety of intellectual backgrounds and are specialists with different, but complementary technical skills. Not only will this project join these skills, but over the next two years a number of other researchers will be added to the project, strengthening, among other aspects, its international component and extending to the training of young scientists in the US.