2025 Simons Collaboration on the Localization of Waves Annual Meeting

The sixth annual meeting of the Localization of Waves Collaboration sponsored by the Simons Foundation hosted about 100 world-class scientists in New York City to discuss their work in understanding and exploiting the localization of waves brought about by a disordered environment or complex geometry, and related wave behaviors. Terence Tao (UCLA) kicked off the presentations with a talk entitled “Delocalization of Eigenvectors of Random Matrices – a Survey,” in which he presented a survey on the recent advances in the field of random matrices. Collaboration Director, Svitlana Mayboroda (ETH-UMN) followed with a talk presenting the “Overview of the Collaboration Progress and Future Directions,” highlighting the successes of the Collaboration’s research during the six years of its existence. Collaboration PI and Nobel Prize winner Alain Aspect (Institut d’Optique Graduate School, IOGS) then gave a talk on “Anderson Localization of Ultracold Atoms in Laser Speckle Disorder” in which he presented breakthrough experiments which achieved for the first time accurate measurements of the Anderson mobility edge in 3D Bose-Einstein condensates of cold atoms. Alessio Figalli (ETH) presented a talk on “Free Boundary Regularity in Obstacle Problems,” detailing the mathematical challenges posed by problems such as the Stefan problem. Concluding the speakers on the first day was Arjun Ashoka (Cambridge) with a speech on “Exploring Disorder Across Time and Length Scales in Semiconductors,” detailing cutting-edge experiments aiming at unveiling the dynamics of inorganic and organic semiconductors at the femtosecond and even attosecond time scale. Day one concluded with PI and speaker dinner at restaurant Portale.

Day two began with a talk by Michael Berry (University of Bristol) on “Wave Trajectories and their Singularities: Madelung, de Broglie, Newton,” in which he discussed some fascinating properties of wave structures emerging when observing them through the prism of semiclassical trajectories of rays. Following was Max Engelstein (UMN) who presented a talk entitled “Robin Green Functions and the Shape of Lungs,” presenting mathematical properties of the Robin harmonic measure at a fractal or prefractal boundary, and its implication on the working of the human lung. The annual meeting concluded with a stimulating presentation by David Spergel (Princeton), “Flames, Plankton Blooms, Clouds, Supernovae and Galactic Winds: How Dynamic Shapes Geometry and Geometry Reveals Dynamics,” describing the challenges and questions posed by the complexity of structures generated by turbulent flows at cosmic scales.

The annual meeting allowed for the Collaboration to again gather and discuss our goals and objectives as we continue our wide reaching and highly impactful studies.

• Svitlana Mayboroda and Marcel Filoche will investigate the mathematical properties of eigenfunctions of the Schrödinger equation in the Wigner-Weyl formalism in random and complex potentials.
• Guy David, Svitlana Mayboroda, Doug Arnold, David Jerison, and Marcel Filoche will develop study the structure of the mobility edge of the Anderson transition in different graphs, such as the Bethe lattice or small-world networks.
• James Speck, together with Claude Weisbuch, will explore the role of impurity disorder and V-defects on the structure of currents in LEDs.
• Marcel Filoche, together with Claude Weisbuch and James Speck, will develop a model of quantum transport in disordered semiconductors to assess carrier mobility in quantum-based drift-diffusion models.
• Alain Aspect, together with Svitlana Mayboroda and Marcel Filoche, will explore the spatial structures of Bose-Einstein condensates observed at the Anderson transition.
• Richard Friend will study the role of dynamic disorder in the behavior of perovskite materials.

We are grateful to the Simons Foundation for the opportunity to continue our research into this exciting topic and look forward to sharing even more results of our efforts following the culmination of the Collaboration in the coming year.

Terence Tao
University of California, Los Angeles

Delocalization of Eigenvectors of Random Matrices – a Survey 

We survey a number of techniques that have been successfully used in recent years to establish delocalization results for eigenvalues of various random matrix ensembles.
 

Svitlana Mayboroda
University of Minnesota

Overview of the Collaboration Progress and Future Directions
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Alain Aspect
Paris-Saclay Institut d’Optique Graduate School

Anderson Localization of Ultracold Atoms in Laser Speckle Disorder
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The observation of 1D localization of ultra-cold atoms in a correlated disorder produced by laser speckle allowed us to observe an effective mobility edge in excellent agreement with theoretical calculations. Beyond qualitative observations, similar experiments in 3D and comparison to theory raise major problems. I will describe our experimental efforts for modifying our experiment to be able to release in the disorder atoms with well-defined energies. I will then present the first success with the improved experimental scheme. Firstly, we achieve direct measurements of the mobility edge with an unprecedented accuracy. Moreover, when releasing atoms with a narrow energy distribution below the mobility edge, in the localized regime, we observe patterns that we are tempted to interpret as localized wave-functions. This opens the perspective to directly test the landscape theoretical description, when we can engineer precisely our disorder, a goal that seems within our reach.
 

Alessio Figalli
ETH Zurich

Free Boundary Regularity in Obstacle Problems

The obstacle problem arises naturally when studying an elastic membrane constrained by contact with an object (the obstacle). This classical problem has captivated mathematicians for over 60 years. In this talk, we will first review the fundamental theory of obstacle problems and then discuss recent developments in the regularity theory of free boundaries.
 

Arjun Ashoka
University of Cambridge

Exploring Disorder Across Time and Length Scales in Semiconductors
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Conventional semiconductors have historically been idealized as perfectly periodic structures where Bloch’s theorem has been applied with great success. However, in reality, condensed matter systems often deviate from this periodic framework across several time and length scales, ranging from static, long-range polycrystalline disorder to dynamic, short-range lattice vibrations. I will demonstrate that long-range static symmetry disorder can lead to the unexpected formation of spin domains in polycrystalline semiconductors, and while short-range static disorder can localize charges, it is often dominated by dynamic disorder originating from the ever-fluctuating crystal lattice background. I will discuss how this dynamic disorder can set the ultimate bounds on the efficiencies of semiconductor devices and demonstrate our work on capturing these effects in established and emerging material systems. Finally, I will lay out a new experimental scheme to capture dynamic disorder on the native phonon timescales in material systems.
 

Michael Berry
University of Bristol

Wave Trajectories and Their Singularities: Madelung, de Broglie, Newton
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The wave counterparts of classical particle paths and geometrical-optics rays are families of trajectories — patterns of streamlines — modified by a ‘quantum potential.’ Wave interference corresponds to undulations in these trajectories, as envisaged by Isaac Newton. Streamline patterns are dominated by singularities at wave vortices and stagnation points. The local momentum (phase gradient of the wave), can exceed the values classically allowed. Regions of such ‘superoscillations’ are bounded by manifolds where the quantum potential is zero. Some classical ‘curl forces’ — not the gradient of a potential — are associated with Hamiltonians (dispersion relations) anisotropic in momentum components, with unusual group velocity field singularities, and eigenfunctions with unfamiliar classical counterparts. For simple dispersion relations, some singularities coincide; for general cases, they are separate.
 

Max Engelstein
University of Minnesota

Robin Green Functions and the Shape of Lungs

Inspired by understanding the shape of mammalian lungs, we investigate the boundary behavior of diffusions with partially reflective (Robin) boundary conditions in rough domains. We show essentially that Robin solutions behave like the solution with completely absorbing (Dirichlet) boundary conditions but averaged at a scale which depends on the probability of reflection.

This observation means that the Robin solutions exhibit drastically different behavior from the Dirichlet solutions. It also allows us to rigorously prove observed and numerically simulated properties of lungs, including their self-similarity dimension and how oxygen absorption changes with the reflection parameter.

This is (ongoing) joint work with G. David, S. Decio, M. Filoche, S. Mayboroda and M. Michetti.
 

David Spergel
The Simons Foundation

Flames, Plankton Blooms, Clouds, Supernovae and Galactic Winds: How Dynamics Shapes Geometry and Geometry Reveals Dynamics

I will discuss a project to explore the behavior of passive and active tracers in turbulent flows. If the turbulent flow has scaling law behavior over a wide range of scales, then we suspect that the geometry of the level of the scalar fields adverted by the flow will also show ADS-like behavior. The dimension of the ADS layer will depend on the ratio of the Kolmogorov timescale (the turnover time of the smallest eddy) to the reaction time for tracer. Thus, observations of the geometry of distant astronomical objects can reveal the underlying physics on scales far smaller than those directly probed by our telescopes. I will present a number of preliminary results and outline the vision for our future work.