## On the number of level sets of smooth Gaussian fields

### With Dmitry Belyaev (Oxford)

# 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.

- Speaker: Dmitry Belyaev (Oxford)
- Tuesday 29 November 2022, 14:00–15:00
- Venue: MR12, Centre for Mathematical Sciences.
- Series: Probability; organiser: Perla Sousi.