2026 Simons Collaboration on the Localization of Waves Annual Meeting

The seventh 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.

The meeting was opened by Collaboration director Svitlana Mayboroda (ETH Zurich–IAS–University of Minnesota), with a presentation entitled “Geometry: The Great and Powerful.” In this talk, she provided a broad overview of the respective roles played by geometry, complexity, and disorder in solutions to wave equations and, more generally, to partial differential equations describing convective or diffusive transport. Douglas Stone (Yale University) followed with a lecture entitled “Controlling and Functionalizing Multiple Scattering in Complex Geometries,” in which he presented theoretical, numerical, and experimental work aimed at manipulating scattering in open systems in order to deliberately engineer and functionalize their inputs and outputs.

Next, Jill Pipher (Brown University) reviewed, in her talk “Regularity of Solutions to Elliptic and Parabolic Operators,” the major developments that have established connections between the geometric properties of domains and the structure of solutions to parabolic partial differential equations defined on them. The meeting then continued with a presentation “Electron Emission from Active Semiconductor Devices: Imaging Hot Electrons and Their Origins” by PI Jim Speck (University of California, Santa Barbara), who discussed recent major advances in LED efficiency. In particular, he described how the long-standing “green gap” problem has been overcome by exploiting defects and dislocations that concentrate charge carriers within the emitting regions. The first day concluded with a thought-provoking lecture by Javier Gómez-Serrano (Brown University), entitled “Modern Mathematics in the Age of AI,” illustrating how artificial intelligence is now helping to drive progress in mathematics and, in some cases, actively contributing to the resolution of open problems.

The second day opened with a presentation by PI Marcel Filoche (ESPCI Paris), who reported on recent experimental work in his talk “Water Waves in Hyperuniform and Fractal Structures.” He focused on the scattering of surface water waves by disordered, hyperuniform, and fractal structures, and the first observation ever of scattering by Fourier quasicrystals. Thierry Giamarchi (University of Geneva) then turned to the physics of interacting quantum many-body waves in his presentation “Waves, Disorder, and Interactions,” highlighting how the presence of interactions dramatically complicates the understanding of such systems. The meeting concluded with a lecture by Jaume de Dios Pont (New York University), entitled “Some Extreme Regimes of the Laplace Operator.” His talk focused in particular on conjectures related to the famous “hot spot conjecture,” which asks whether the maximum of a Laplacian eigenfunction with Neumann boundary conditions must necessarily be located on the boundary of the domain.

We are deeply grateful to the Simons Foundation for having given us the opportunity to research into many exciting topics during the duration of our Collaboration.

Jaume de Dios Pont
New York University

Some Extreme Regimes of the Laplace Operator

The Laplace operator plays a central role in the mathematical description of many physical phenomena, such as quantum mechanics, wave propagation, or diffusion processes. This lets us, in many classical settings, use physical intuition to interpret the spectral properties of the Laplace operator on a fixed domain by considering a constant coefficient version of the physical phenomenon at hand.
This talk takes a complementary point of view. We study the behavior of the first few eigenfunctions of the Laplace operator in extreme geometric or asymptotic regimes. In the limit, the operator itself undergoes a qualitative change, recovering variable coefficient physical equations. Depending on the regime, the limiting behavior may be described by an effective Schrödinger operator, a drift–diffusion equation, or a heat equation with an emergent time variable. Such effects arise, for instance, in high dimensional domains, highly perforated or sieve-type domains, as well as in situations where geometric constraints enforce strong localization.
Several classical problems will serve as guiding examples, including questions related to the hot spots and KLS conjectures, homogenization for perforated domains, and rigidity phenomena for eigenfunctions of convex sets. Although these topics are not governed by a single unifying mechanism, they share a common feature: informative spectral behavior becomes visible only in limiting regimes where an effective operator emerges. The goal is to illustrate how pushing the Laplacian into such extreme settings sheds light on localization, geometry, and the structure of low-energy eigenfunctions.
 

Marcel Filoche
ESPCI Paris – PSL University

Water Waves in Hyperuniform and Fractal Structures
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Surface water waves provide a unique experimental platform for probing wave transport in two-dimensional structures by offering the rare capability of direct, spatially, and time-resolved measurements of the full complex wavefield, both in amplitude and phase. Moreover, surface waves allow exploration across a broad wavelength range, enabling the investigation of complex geometries and scattering processes and direct access to the underlying structure of energy transport. In this talk, we present an experimental setup using water waves to test various types of disorder, from hyperuniform to fractal. We report the first experimental observation of a transition from a scattering to a non-scattering regime for water waves propagating through a stealthy hyperuniform (SHU) disorder. One key feature of this platform is that water acts as an absorbing background medium, introducing dissipation that competes with correlation-induced transparency. This dissipation is also enhanced by the introduction of pre-fractal structures, enabling us to probe not only the ideal theoretical predictions for several types of disorder, but also their robustness in realistic, lossy environments.
 

Thierry Giamarchi
University of Geneva

Waves, Disorder, and Interactions

As discovered in the seminal paper of P. W. Anderson in 1958 when an equation such as the Schrödinger equation (and other related wave equations) is subjected to a random potential, the nature of the solutions changes drastically going from plane waves to localizes states. An important question is what happens to this phenomenon when instead of looking at the properties of one single particle one wants to deal with a large number of interacting quantum particles, as is relevant for several experimental realizations, both in cold atomic gases and in condensed matter.

This talk provides an overview of this class of phenomena, with questions ranging from the phases that can be reached in such interacting disordered systems to the consequences for the transport properties of such systems, and finally the delicate question of the role of the temperature, in presence or absence of a thermal bath. I will also discuss what happens for other classes of potentials than the plain disorder, such as quasiperiodic potentials, or colored noise, both from a theoretical perspective but also in contact with recent experiments in cold atomic gases. I will point to the challenges in the field.

Disclaimer: Since it is a talk given by a physicist, there will unfortunately be no theorems but a set of “unproven” results, some of which could perhaps be called “conjectures,” and which hopefully will stimulate the curiosity of a more rigorously inclined audience.
 

Javier Gómez-Serrano
Brown University

“Modern Mathematics” in the Age of AI
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In this talk, I will explain several recent and ongoing results combining machine-learning techniques and more traditional mathematics. The overarching theme is the interplay between AI and human expertise in order to discover new solutions of certain mathematical problems, leading sometimes to mathematics at scale and even automated, end-to-end rigorous mathematical proofs.
 

Svitlana Mayboroda
University of Minnesota

Overview of the Collaboration Progress and Future Directions
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The talk will be devoted to an overview of the progress of the Collaboration to date, along with a discussion of open problems and future directions.
 

Jill Pipher
Brown University

Regularity of Solutions to Elliptic and Parabolic Operators
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This lecture will trace some of the historical milestones, up through recent developments, for this field at the interface of harmonic analysis, geometric measure theory, and PDE. Between 1957 and 1961, several seemingly unrelated mathematical discoveries appeared: Carleson measures in analytic interpolation, De Giorgi-Nash regularity of solutions to elliptic and parabolic divergence form equations with bounded measurable coefficients, and the John-Nirenberg space of bounded mean oscillation (BMO) functions. (Published simultaneously, Moser used BMO in his approach to regularity of solutions.) In 1971, C. Fefferman illuminated the profound connections between harmonic functions and Carleson measures and BMO. This opened up a theory of boundary value problems aimed at quantifying the connections between the geometry of domains, the regularity of coefficients of elliptic/parabolic operators, and the behavior of solutions. Evolving over a half century, this theory has found new direction and significance in the work of the Localization of Waves Collaboration.
 

Jim Speck
University of California Santa Barbara

Electron Emission from Active Semiconductor Devices: Imaging Hot Electrons and Their Origins
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Semiconductor materials and devices serve as the foundation of nearly all electronic systems that power our daily lives. While the physics in these semiconductor systems is well described by theory, little has been done to experimentally study the carrier physics in active devices.

This talk will focus on our recent progress on electron emission microscopy (EEM) and spectroscopy (EES) of III-N semiconductor devices. EEM images the internally generated electrons emitted from active devices, allowing us directly “see” the electrons in these semiconductor devices in-operando. EES measurements yield the energies of these emitted electrons, giving us insights into their physical origins.

Studies of current spreading on p-n diodes and carrier injection in LEDs show phenomenal agreement with theory and simulations. On the other hand, studies of the recombination mechanisms in p-n diodes disagree with conventionally employed defect-assisted recombination models.
 

Douglas Stone
Yale University

Controlling and Functionalizing Multiple Scattering in Complex Geometries
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Harnessing interference to control wave propagation in multiple scattering geometries is a fundamental challenge in both classical and quantum physics. With the advent of new tools for wavefront shaping, it has become possible to optimize target functions over a space of possible coherent input fields, but there was little mathematical understanding of the achievable performance, leading to a reliance on black-box optimization techniques. We have shown that it is possible to formulate a large class of useful scattering target functions as non-Hermitian eigenvalue problems, a first example being coherent perfect absorption, or time-reversed lasing. This implies that a class of optimal solutions exist as spectra in the complex frequency plane and can lead to physical steady-state solutions by tuning system parameters to move one such eigenfrequency to the real axis. More highly constrained solutions, e.g., demultiplexing of a multi-frequency input, are achieved by generating coincidences of these eigenfrequencies. We show in simulations and experiments that a “chaotic cavity” containing tunable scattering elements is a robust platform for realizing this scenario. The general mathematical framework presented applies to all the linear classical and quantum wave equations of physics.

Watch a playlist of all presentations from this meeting here.