Talks

Turbulence generated by nonlinear interactions between waves is the mechanism behind many physical phenomena, e.g., surface wave turbulence on the ocean, Alfvén wave turbulence, and drift wave turbulence in fusion plasma. The canonical theoretical approach is to write down a closed, nonlinear integral equation for the evolution of the number operator (amplitude squared of the wave function) in Fourier space. This equation emerges from the quasi-normal approximation. The scaling laws for energy and the number operator can be derived from the kinetic equation. Recently, under certain conditions (one of which is a dimension greater than or equal to three), the kinetic equation has been rigorously derived without resorting to the quasi-normal assumption. This work was awarded the Fields Medal in 2026. However, most wave turbulence problems are relevant in two dimensions, and several other assumptions made in the rigorous derivation of the kinetic equations do not hold in practice. Furthermore, there is weak support for the scaling predictions obtained from the kinetic equation from experiments and simulations because in most cases the scaling range is too small for reliable determination of scaling exponents.

We study a one-dimensional model of wave turbulence (Majda-McLaughlin-Tabak) using the dynamic renormalization group and massive direct numerical simulations (2^22 grid points). Our simulations with three decades of scaling range show that the predictions of the kinetic equation do not hold. Crucially, higher-order (even) moments of the wave function show scaling with exponents that depend nonlinearly on the order, i.e., we obtain multiscaling. This is the only model that shows multiscaling in one dimension. Our theory cannot yet capture multiscaling.
 

This set of lectures will introduce concepts in nonlinear dynamics and chaos, illustrated by interdisciplinary examples, ranging from mechanical vibrations to biological rhythms. The topics will include differential equations, bifurcations, phase plane analysis of flows on lines and circles, introduction to characteristics of chaos, iterated maps, routes to chaos, and introductory concepts in renormalization, fractals, multifractals and strange attractors.

Equilibrium statistical mechanics is built on thermal fluctuations and energy conservation. Many systems of current interest, however, including granular materials, dense suspensions, and amorphous solids, are essentially athermal, with dynamics governed instead by mechanical constraints, external driving, and dissipation.
This course provides an introduction to the statistical physics of athermal matter, focusing on how microscopic constraints give rise to macroscopic mechanical behavior. We begin with the jamming transition, isostaticity, and marginal stability, before discussing the vibrational spectrum of amorphous solids, soft modes, and the emergence of anomalous elasticity. We then examine the scaling of elastic moduli, long-range stress correlations, and continuum descriptions of disordered solids. Finally, we turn to dynamics, including yielding, plasticity, dense suspension rheology, and hydrodynamic descriptions of flowing athermal systems, highlighting some of the central open problems in the field.