On the number of level sets of smooth Gaussian fields

The number of zeroes or, more generally, level crossings of a Gaussian process is a classical subject that goes back to the works of Kac and Rice who studied zeroes of random polynomials. The number of zeroes or level crossings has two natural generalizations in higher dimensions. One can either look at the size of the level set or the number of connected components. The surface area of a level set could be computed in a similar way using Kac-Rice formulas. On the other hand, the number of the connected components is a ‘non-local’ quantity which is notoriously hard to work with. The law of large numbers has been established by Nazarov and Sodin about ten years ago. In this talk, we will briefly discuss their work and then discuss the recent progress in estimating the variance and deriving the central limit theorem. The talk is based on joint work with M. McAuley and S. Muirhead.