Over the last decade there have been substantial theoretical advances in the geometry of random fields, mainly inspired by applications to human brain mapping. The purpose of this talk is to present these to the astrophysics community, and to apply them to CMB from the first-year WMAP, and galaxy density from the first-year SDSS. The new results all concern the genus or Euler characteristic (EC) of excursion sets of random fields. A decade ago the classic Gaussian results were extended to non-Gaussian fields such as chi^2, t and F statistic random fields, increasing the number of 'toy' models for which exact expected EC is known. An interesting case is the chi^2 field with two degrees of freedom, whose excursion set forms closed strings that can in fact be linked or knotted with positive probability. More recently results have become available for random fields whose values are the sample correlation at every pair of points, and for random fields of other multivariate test statistics such as Hotelling's T^2 and Roy's maximum root. Results are also available for conjunctions or intersections of independent excursion sets, that is, the minimum of an arbitrary number of any of the above random fields. Astrophysics data is often smoothed with a kernel of arbitrary width before analysis. We vary the amount of smoothing, adding an extra parameter to the random field, so-called 'scale space'. Once again we can find exact EC results for such fields. All of these results are easily extended to other Minkowski functionals such as caliper diameter and surface area. Finally, boundary corrections are available for when the search region is finite (such as CMB on the sphere) rather than infinite (such as galaxy density).
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