Gaussian random fields are fully characterized by their two-point correlation function or power-spectrum. Local properties, such as the density of maxima, minima and saddle points can be analytically evaluated. Non-local properties are known to be more complicated to evaluate. A good example of a non-local statistics which can be evaluated exactly is the Euler characteristic (Bardeen et al.). In this project we analytically study the non-local statistics of disconnected components, and loops (the homology) in super level set filtrations of two-dimensional Gaussian random fields.

This is work in collaboration with Rien van de Weygaert and Matti van Engelen.

__Job Feldbrugge__, Matti van Engelen, Rien van de Weygaert, Pratyush Pranav, Gert Vegter.
** Stochastic Homology of Gaussian vs. non-Gaussian Random Fields: Graphs towards Betti Numbers and Persistence Diagrams**. 2019.
[arXiv][pdf]

Pratyush Pranav, Rien van de Weygaert, Gert Vegter, Bernard J. T. Jones,
Robert J. Adler, __Job Feldbrugge__, Changbom Park, Thomas Buchert,
Michael Kerber.
** Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals**. 2018.
[arXiv][pdf]

R. van de Weygaert, G. Vegter, H. Edelsbrunner, B. Jones, P. Pranav, C. Park, W. Hellwing, B. Eldering, N. Kruithof, E. Bos, J. Hidding, __Job Feldbrugge__, E. ten Have, M. van Engelen, M. Caroli, M. Teillaud
** Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web**. 2013.
[arXiv][pdf]

Bachelor thesis advised by Rien van de Weygaert, Gert Vegter, and Elisabetta Pallant [pdf].