Svitlana Mayboroda, University of Minnesota and ETH Zurich
Marcel Filoche, École Supérieure de Physique et Chimie Industrielle
The 2024 Annual Meeting of the Simons Collaboration on Localization of Waves will bring together top mathematicians and physicists who work on understanding and manipulating the behavior of waves in disordered media or complex geometry, with a particular focus on localization phenomena.
The two-day meeting will feature presentations of recent advances in the mathematics, physics, and applications of localization. These include new results on the geometric structure of random waves, theoretical advances in Anderson localization, and groundbreaking experiments in systems of cold atoms.
The meeting will also be an opportunity for all participants to engage in open discussion, exchange ideas and make new connections.
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Svitlana Mayboroda | Brownian Travelers & Boundary Complexity 10:30 AM BREAK 11:00 AM Bart van Tiggelen | The Theory of White Paint Revisited 12:00 PM LUNCH 1:00 PM Hugo Duminil-Copin | A New Way of Looking at High-Dimensional Lattice Models 2:00 PM BREAK 2:30 PM Douglas Arnold | What the @#$! is Cohomology Doing in Numerical Analysis?! 3:30 PM BREAK 4:00 PM David Huse | Many-Body Localization 5:00 PM DAY ONE CONCLUDES
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Horng-Tzer Yau | Random Band Matrices, Localization and Quantum Unique Ergodicity 10:30 AM BREAK 11:00 AM Marcel Filoche | The Structure of the Anderson Transition 12:00 PM LUNCH 1:00 PM Jeffrey Ovall | Computational Tools for Exploring Eigenvector Localization 2:00 PM MEETING CONCLUDES
University of Minnesota and ETH Zurich
Brownian Travelers & Boundary Complexity
Svitlana Mayboroda will discuss how the Brownian travelers see irregular, disordered, possibly higher dimensional boundaries, and how geometric complexity, in nature or in engineering, enhances robustness and efficiency of the underlying physical systems. From the point of view of mathematics, these are notoriously difficult problems of the structure and support of the harmonic measure, as well as regularity of the emerging free boundaries. Applications in physics range from the construction of noise abatement walls to transport in lung to heterogeneous catalysis. In particular, we prove that contrary to the conjectures put forward by physicists and contrary to the “classical” Dirichlet scenario, the harmonic measure associated to a partially reflective Brownian motion is absolutely continuous with respect to the Hausdorff measure on any boundary.
Bart van Tiggelen
Université Grenoble Alpes
The Theory of White Paint Revisited
Numerical simulations have demonstrated the absence of an Anderson transition when light propagates in a dense 3D medium filled with electric dipoles. They have no easy explanation, and all existing theories predict a mobility edge when the Ioffe-Regel parameter (a product of wavenumber and mean free path) is of the order of one. Some crucial element must have been overlooked. Longitudinal electric waves are well known in classical and quantum electromagnetism, but their role has so far been underestimated in radiative transfer. The reason is that, alone, they cannot propagate because they do not create a magnetic field and therefore no Poynting vector, as is the case for transversely polarized waves. We have reconsidered the transport theory of electromagnetic waves in 3D random media and found that the interference of longitudinal and transverse waves creates a second channel in transport. Because the two channels are coupled by scattering, it is much more difficult to close both simultaneously and to have Anderson localization.
Université de Genève
A New Way of Looking at High-Dimensional Lattice Models
Embarking on an exploration of high-dimensional lattice models, this presentation delves into contemporary advancements within the realm of percolation, self-avoiding walks, the Ising model, and the XY model. Our approach diverges from traditional lace-expansion and renormalization techniques, opting for an innovative perspective that elucidates mean-field behavior. The crux of our methodology lies in an examination of the random-walk representation inherent in these intricate models.
University of Minnesota
What the @#$! is Cohomology Doing in Numerical Analysis?!
As the name suggests, numerical analysis — the study of computational algorithms to solve mathematical problems, such as systems of differential equations — has traditionally been viewed mostly as a branch of analysis. Geometry, topology and algebra played little role. However, in the last decade or so, things have changed. The recent literature on numerical analysis is replete with papers using concepts that are new to the subject, say, symplectic differential forms or de Rham cohomology or Hodge theory. In this talk, Douglas Arnold will discuss some examples of this phenomenon, especially the finite element exterior calculus. We shall see why these new ideas arise naturally in numerical analysis and how they contribute.
Many-body localization (MBL) is Anderson localization of many interacting quantum degrees of freedom in highly-excited states at conditions that correspond to a nonzero entropy density at thermal equilibrium. The opposite of MBL is thermalization, where the isolated quantum many-body system successfully acts as a thermal bath for itself, bringing all of its small subsystems to thermal equilibrium with each other via the unitary quantum dynamics of the closed system. MBL, unlike wave or single-particle localization, does not appear to have a weak localization regime.
For systems with short-range interactions, the transition from thermalization to MBL occurs in two stages as the interactions are reduced: First is a smooth crossover to a “glassy” prethermal MBL regime, where the thermalization time of a large system becomes extremely large but not infinite. Then, at still weaker interaction is the dynamical phase transition in to the MBL phase, which in some cases occurs at a strength of interactions that is so small that it is thermodynamically insignificant in the limit of large systems, even though it has strong long-time dynamical effects.
The Structure of the Anderson Transition
The structure of the phase diagram of eigenfunction localization for tight-binding Hamiltonians with random independently and identically distributed (i.i.d.)P disorder (à la Anderson) in the energy-disorder plane is well known: below dimension d=2, all eigenfunctions are localized for non-vanishing disorder, and above d=2, a delocalized phase appears separated from the localized phase by a transition line called the “mobility edge,” predicted by the so-called self-consistent theory of localization in the case of uniform disorder. Marcel Filoche will show that behind this simple description, there is in fact a more complicated structure emerging already at a low dimension and will explore the various mechanisms at work in the localization/delocalization transition.
Portland State University
Computational Tools for Exploring Eigenvector Localization
We present an algorithm for determining all eigenpairs of the magnetic Schrödinger operator whose eigenvalues lie within a user-specified range, and whose eigenvectors are sufficiently concentrated within a user-specified subdomain (not just “ground states”) or certifying that no such eigenpairs exist. The algorithm is based on a relatively compact perturbation of the magnetic Schrödinger operator that is designed to highlight only strong candidates for eigenpairs satisfying the given criteria. Heuristic, theoretical and computational support will be provided for the algorithm. Although the theoretical results apply to relatively generic magnetic Schrödinger operators, the numerical illustrations will focus on the two “extreme” cases: the standard Schrödinger operator and the magnetic Laplacian operator. In contrast to the standard Schrödinger operator, for which a rich mathematical theory based on the localization landscape provides a fairly detailed understanding of localization in the lower part of the spectrum, not much is known about the mechanisms driving localization for the magnetic Laplacian, and we provide some theoretical and computational insight.
Participation & Funding
Participation in the meeting falls into the following four categories. An individual’s participation category is communicated via their letter of invitation.
Group A – PIs and Speakers
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Business-class or premium economy airfare will be booked for all flights over five hours.
Group B – Out-of-town Participants
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Economy-class airfare will be booked for all flights.
Group C – Local Participants
Individuals in Group C are considered local and will not receive financial support, but are encouraged to enjoy all conference-hosted meals.
Group D – Remote Participants
Individuals in Group D will participate in the meeting remotely. Please register at the link above and a remote participation link will be sent to you approximately two weeks prior to the meeting.
Travel & Hotel
Air and Rail
For individuals in Groups A and B the foundation will arrange and pay for round-trip travel from their home city to the conference.
All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency.
Travel specifications, including preferred airline, will be accommodated provided that these specifications are reasonable and within budget.
Travel arrangements not booked through the preferred agency, including triangle trips and routing/preferred airlines outside budget, must be pre-approved by the Simons Foundation and a reimbursement quote must be obtained through the foundation’s travel agency.
Personal & Rental Cars
Personal car and rental trips over 250 miles each way require prior approval from the Simons Foundation via email.
Rental cars must be pre-approved by the Simons Foundation.
The James NoMad Hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation.
Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The James NoMad Hotel. Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link above.
The James NoMad Hotel
22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)
For driving directions to The James NoMad, please click here.
Individuals in Groups A & B will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.
Expenses accrued as a result of meetings not directly related to the Simons Foundation-hosted meeting (a satellite collaboration meeting held at another institution, for example) will not be reimbursed by the Simons Foundation and should be paid by other sources.
Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.
The daily meal limit is $125 and itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses.
- Meals taken on travel days are reimbursable.
- Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner) are not reimbursable.
- If a meal was not provided on a meeting day, dinner for example, that expense is reimbursable.
- Meals taken on days not associated with Simons Foundation-coordinated events are not reimbursable.
- Minibar expenses are not reimbursable
- Meal expenses for a non-foundation guest are not reimbursable.
Group meals consisting of fellow meeting participants paid by a single person will be reimbursed up to $65 per person per meal and the amount will count towards each individual’s $125 daily meal limit.
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport) as well as local transportation. While in NYC, individuals are encouraged to use public transportation and not use taxi, Uber or Lyft services.
Attendance & Building Protocols
In-person participants and speakers are expected to attend all meeting days. Partial participation is permitted so long as the individual fully attends the first day, which is typically Thursday for two-day meetings. Participants receiving hotel and travel support wishing to arrive on meeting days which conclude at 2:00 PM will be asked to attend remotely.
Individuals accessing Simons Foundation and Flatiron Institute buildings must be fully vaccinated against COVID-19.
Entry & Building Access
Upon arrival, guests will be required to show their photo ID to enter the Simons Foundation and Flatiron Institute buildings. After checking-in at the meeting reception desk, guests will be able to show their meeting name badge to re-enter the building. If you forget your name badge, you will need to provide your photo ID.
The Simons Foundation and Flatiron Institute buildings are not considered “open campuses” and meeting participants will only have access to the spaces in which the meeting will take place. All other areas are off limits without prior approval.
If you require a private space to conduct a phone call or remote meeting, please contact your meeting manager at least 48-hours ahead of time so that they may book a space for you within the foundation’s room reservation system.
Guests & Children
Meeting participants are required to give 24 hour advance notice of any guests meeting them at the Simons Foundation either before or after the meeting. Outside guests are discouraged from joining meeting activities, including meals.
With the exception of Simons Foundation and Flatiron Institute staff, ad hoc meeting participants who did not receive a meeting invitation directly from the Simons Foundation are not permitted.
Children under the age of 18 are not permitted to attend meetings at the Simons Foundation. Furthermore, the Simons Foundation does not provide childcare facilities or support of any kind. Special accommodations will be made for nursing parents.