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Zero sets of random functions

Periodic Reporting for period 3 - RandomZeroSets (Zero sets of random functions)

Reporting period: 2019-10-01 to 2021-03-31

The project is focused on zero sets of random functions. This is a rapidly growing area that pertains to
pure mathematics (which is a fundamental science) and lies at the crossroads of analysis, probability theory, and mathematical physics. Various instances of zero sets of random functions have been used to model phenomena in quantum chaos, real algebraic geometry, theory of entire functions, and theory of random point processes.

The challenging problems addressed in the proposal connect classical fields of mathematics with such rapidly developing disciplines as spectral geometry, statistical topology and percolation theory.
We have investigated a variety of topics, from zero sets of harmonic functions and random spherical harmonics to rigidity of random processes.
Our work led us to answers to several long-standing open problems (e.g. Nadirashavili and Nevai conjectures) and to several quite unexpected findings concerning behavior of discrete harmonic functions and of linear combination of Laplace eigenfunctions.
We plan to proceed with this line of research till the end of the project, and hope that the second half of the project period will be at least as successful as was the first half. One of our priorities is to prepare for publication our work with Fedor Nazarov which provides the first rigorous justification of the physical heuristics suggested by Bogomolny and Schmid on the relation between statistical topology of zero sets of smooth Gaussian ensembles and loop models.