Periodic Reporting for period 5 - RandomZeroSets (Zero sets of random functions)
Período documentado: 2022-10-01 hasta 2023-09-30
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 (analysis, probability theory, mathematical physics, and number theory) with such rapidly developing disciplines as spectral geometry, statistical topology, percolation theory.
We have investigated a variety of topics, from zero sets of random spherical harmonic, of discrete harmonic functions and of linear combinations of Laplacian eigenfunctions, to rigidity of random processes, zeroes of Taylor series with random and psuedo-random coefficients,
and measurable entire functions.
and measurable entire functions.
Our work led us to answers to several long-standing open problems (e.g. Nadirashavili and Nevai conjectures) and to several quite unexpected discoveries concerning behavior of discrete harmonic functions and of linear combination of Laplace eigenfunctions.