Mathematics Colloquium - Recent progress in understanding the nodal structures of random fields
Speaker : Igor Wigman from King's College London - Igor is a leading expert in the study of zero sets of random Gaussian functions and using number-theoretic methods towards it. He also is the recipient of an ERC Standard Grant, one of the most prestigious sources of funding in the European academic system.
Abstract: The nodal set of a nice function defined on a smooth manifold or the Euclidean space is its zero set. The study of nodal sets of Gaussian random fields has positioned itself in the heart of several disciplines, including probability theory and spectral geometry, and, more recently, it has exhibited connections to number theory. We are interested in the asymptotic topology and geometry of the nodal lines in the high energy limit. In the first part of the talk I will give an overview of the classical results in this field, and the related methods. In the second part of the talk I will describe the more recent progress, related to percolation properties of the nodal lines, inspired by the beautiful percolation model due to Bogomolny-Schmit. Finally, I will describe the recent results obtained in a joint work with D. Beliaev and S. Muirhead on the relation between the percolation properties of the nodal sets and their connectivity measures, that were defined and whose existence was established in a joint work with P. Sarnak.
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Ellen LynchMathematics Colloquium - Recent progress in understanding the nodal structures of random fields