On the number of level sets of smooth Gaussian fields - II

Abstract: Level sets of smooth Gaussian fields appear in many areas of mathematics. They have numerous applications outside of mathematics from astrophysics to oceanology to biology. Probably the first significant progress in the study of level lines came in 1940s when Kac and Rice developed formulas that allow to compute the expected number of roots of a random function in 1d. These formulas can be generalised to higher dimension where they allow to compute the expected volume of level sets. Unfortunately, these formulas do not allow to study the number of level lines since it is a non-local quantity which can not be written as a Kac-Rice-type integral formula. In this talk we will discuss recent progress in the study of the number of level lines for smooth Gaussian fields.

In the first part of the talk D. Belyaev will give a gentle introduction to the area and give a broad overview of recent results. In particular, he will describe how the number of level lines depends on the level. This allows to show that the expected number of lines depends (almost) smoothly on the level and allows to give a low bound on the fluctuation of the number of level lines. In the second part S. Muirhead will explain in more details the ideas behind these results and will sketch the proof of the variance bound.The talk is based on a series of papers by D. Belyaev, M. McAuley and S. Muirhead.

probability

Audience: advanced learners

Bangalore Probability Seminar

Series comments: The link to zoom meeting can be found on the seminar's google calendar - www.isibang.ac.in/~d.yogesh/BPS.html