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Nodal Lines

Final Report Summary - NODAL (Nodal Lines)

This project was inspired by the experiments conducted by the physicist and musician Ernst Chladni in the 18th century. Nodal lines (also referred to as "Chladni Plates" or "Chladni Modes") describe sets that remain stationary during membrane vibrations, hence their importance in such diverse areas as musical instruments industry, mechanical structures, earthquake study and other fields. In the duration of this ERC project these were addressed with mathematical rigour.

One of the major accomplishments of the project is a uncovering a surprising and striking relation between the nodal structures and the arithmetic of lattice points lying on circles, a classical problem within analytic number theory (two squares problem). This allowed for a systematic study of the nodal structures on some particularly important arithmetic dynamical systems, such as the torus or the square, by invoking powerful methods borrowing from Number Theory, complemented by modern techniques within probability theory, such is the Kac-Rice formulae and Wiener chaos decomposition.

Another striking accomplishment of the project is the discovery of a universal law for the distribution of types of nodal domains and their mutual positions, applicable in a wide variety of situations and dynamical systems, occurring both in mathematical physics, probability theory, spectral geometry and complex geometry. It was numerically understood that there exists a phase transition between the 2-dimensional situation and the 3 and higher dimensional cases. This was explained by the critical percolation (Bogomolny-Schmit within the physics literature) governing the nodal structures in 2d as opposed to the higher dimensional case, governed by super-critical percolation, where a single giant component is dominating a vast part of the nodal structures. An important outcome of the project is corroborating this ansatz.