Robert J. Adler

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RANDOM FIELDS AND GEOMETRY published with Springer in 2007, but rather a companion volume, still under production, that gives a simpler version of the theory of the first book as well as lots of applications. You can find the original Random Fields and Geometry on the Springer site. Meanwhile, enjoy what is available of the second volume, and keep in mind(More)
We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process f . Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation.(More)
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the(More)
This is a brief review, in relatively nontechnical terms, of recent rather technical advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric characteristics of excursion sets of random fields, such as their volume, surface area and Euler(More)
We are interested in computing tail probabilities for the maxima of Gaussian random fields. In this paper, we discuss two special cases: random fields defined over a finite number of distinct point and fields with finite Karhunen-Loève expansions. For the first case we propose an importance sampling estimator which yields asymptotically zero relative(More)
over high levels u. For a large class of such random fields we compute the u → ∞ asymptotic joint distribution of the numbers of critical points, of various types, of X in Au, conditional on Au being non-empty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set. In a significant(More)