Invitation Only
Organizer:
Jalal Shatah, New York University
Speakers:
Tristan Buckmaster, New York University
Luc Deike, Princeton University
Gregory Falkovich, Weizmann Institute of Science
Erwin Faou, National Institute for Research in Digital Science and Technology
Nader Masmoudi, New York University
Sylvia Serfaty, New York University
Gigliola Staffilani, Massachusetts Institute of Technology
Stephanie Waterman, University of British Columbia
Meeting Goals:
The annual meeting for the Simons Collaboration in Wave Turbulence Theory will feature a diverse and multidisciplinary body of work that pushes the boundaries of both theory and experiment in fluid mechanics, turbulence, PDE, and computational mathematics, offering new methods and insights across classical and modern approaches.
A more granular description of the featured topics are as follows:
Mean-Field Limits and Modulated Energy Methods
A central theme in this body of work is the rigorous derivation of mean-field limits for systems of particles with singular interactions—notably Coulomb and Riesz types. These systems are governed by gradient flows, conservative flows, and may include stochastic (noisy) effects. The modulated energy method is introduced as a tool to quantify convergence from a discrete particle system to a continuum PDE limit. At the heart of this approach lies a commutator-type functional inequality, which has seen significant recent progress. Global-in-time convergence is also addressed.
Renormalization and Wave Turbulence Theory
Another topic featured at the meeting is the application of renormalization techniques, traditionally used in quantum field theory, to the theory of wave turbulence. This perspective allows for a unified framework across a wide range of nonlinearities—leading to the development of a renormalized kinetic equation. This integro-differential equation can model wave turbulence from weak to strong regimes and supports a new classification of universality classes based on the role of nonlinear interaction enhancement or suppression.
Energy Cascades and Forced Nonlinear Wave Equations
A more elementary but insightful approach is presented to explore energy cascades in forced nonlinear wave equations, including the nonlinear Schrödinger and shallow water equations. By carefully choosing smooth forcing terms, quasi-singular solutions exhibiting power-law decay in Fourier space, whose exponents depend on the system, are constructed. Some stability results are also provided, ensuring that these power-law behaviors persist under perturbations.
Computational and Machine Learning Approaches to PDEs
The meeting will also include results that applies Physics-Informed Neural Networks (PINNs) to discover novel solutions to nonlinear PDEs, including unstable ones that are otherwise difficult to obtain. These computational insights are then translated into rigorous mathematical proofs using computer-assisted methods. While inspired by classical fluid equations (Euler, Navier-Stokes), the methodologies have broad applicability across PDEs.
Boundary Conditions, Energy Spectra & Number Theory
It is common practice among mathematicians and physicists to assume periodic boundary conditions when analyzing the energy spectrum of evolutionary equations. However, an important question arises: how do these boundary conditions affect for example energy transfer in nonlinear Schrödinger equations? Interestingly, this investigation reveals unexpected connections with analytic number theory and leads to the construction of special solutions that display distinct energy transfer behavior.
Understanding Ocean Mixing: Quantifying Turbulence Across Scales
One of the themes presented by experimentalists concerns the ocean mixing processes and how crucial they are in shaping the global distribution of heat, energy, chemical tracers, and marine life, with direct implications for climate dynamics. However, significant uncertainty remains regarding the rates and mechanisms of mixing across a wide range of spatial and temporal scales. Scientists combine traditional approaches with innovative theoretical and observational techniques to quantify turbulence and mixing in the ocean, aiming to uncover the underlying processes that drive these dynamics at both large and small scales.
Nonlinear and Multiscale Fluid Dynamics
There are fundamental problems in fluid dynamics, of relevance to both environmental and industrial applications, that require placing strong emphasis on nonlinear and multiscale systems. These include the statistical behavior of ocean waves, interactions with floating ice sheets, gas exchange driven by surface-breaking waves, spray formation and transport in the atmosphere, and strategies for mitigating oil spills.
To investigate these phenomena, researchers design and conduct complementary laboratory and numerical experiments aimed at developing simplified physical models that capture the key dynamics of each process. Particular attention is given to the study of breaking waves, bubble dynamics in turbulent flows, gas transfer mechanisms, and the generation of spray through bubble bursting.
Past Meetings:
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Thursday, December 4, 2025
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Gigliola Staffilani | Some Mathematically Rigorous Results on Wave Turbulence Theory 10:30 AM BREAK 11:00 AM Nader Masmoudi | Recent Advances in Nonlinear Inviscid Damping 12:00 PM LUNCH 1:00 PM Luc Deike | Broadband Ocean Waves Dynamics and Statistics Involving Wind Forcing and Dissipation by Breaking Through a Multilayer Model 2:00 PM BREAK 2:30 PM Erwan Faou | Quasi-singularities and power laws solutions for PDEs with forcing 3:30 PM BREAK 4:00 PM Tristan Buckmaster | Fluid Singularities, Unstable PDE Solutions and Computer-Assisted Proofs 5:00 PM DAY ONE CONCLUDES Friday, December 5, 2025
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Sylvia Serfaty | Mean-Field Limits by the Modulated Energy Method 10:30 AM BREAK 11:00 AM Stephanie Waterman | Filling in the Map: Understanding Arctic Ocean Mixing Rates, Mechanisms, Space-Time Variability and Decadal-Scale Trends From Ocean Observations 12:00 PM LUNCH 1:00 PM Gregory Falkovich | One Equation to Rule Them All, Weak and Strong 2:00 PM MEETING CONCLUDES -
Tristan Buckmaster
New York UniversityFluid Singularities, Unstable PDE Solutions and Computer-Assisted Proofs
This talk presents recent work on understanding certain solutions of PDE by combining modern mathematics with classical analysis. Machine learning, particularly physics-informed neural networks (PINNs), is being applied to discover new solutions to nonlinear PDEs with high accuracy. A key aspect is the interplay between these methods to uncover the full spectrum of solutions, significantly, unstable solutions that are challenging to find otherwise.
We will also demonstrate how computer-assisted methods can transform these numerical discoveries into rigorous mathematical proofs. While motivated by fluid mechanics equations such as Euler and Navier-Stokes, the methods discussed have broader applicability to other PDEs.
Luc Deike
Princeton UniversityBroadband Ocean Waves Dynamics and Statistics Involving Wind Forcing and Dissipation by Breaking Through a Multilayer Model
The ocean wave statistics is controlled by the balance between energy input by the wind, weakly non-linear interaction between waves and energy dissipation due to breaking. I will discuss a multilayer numerical framework which generalizes the single-layer Saint-Venant system into a multi-layer, non-hydrostatic formulation of the Navier-Stokes equations (Popinet, J. Comput. Phys. 2020; Wu et al., J. Fluid Mech. 2023, 2025) allowing to resolve broad banded wave fields in presence of wave breaking and under wind forcing. I will discuss the resulting wave spectra, wave breaking statistics and upper ocean turbulence for both steady and growing wave fields over extended periods of times (several hundreds of peak wave periods).
Erwan Faou
National Institute for Research in Digital Science and TechnologyQuasi-singularities and power laws solutions for PDEs with forcing
In this talk I will address the problem of singularity and quasi-singularity formation in partial differential equations with smooth forcing and smooth coefficients. In simple models like the nonlinear Schroedinger equation and the shallow water equation, I will show how a smooth forcing can yield to solutions exhibiting quasi singularities. While the forcing terms considered are smooth and well localized in Fourier, these solutions have Fourier spectra decaying at power laws rate depending on the algebraic structure of the nonlinearity of the equation. I will then study the stability of such solutions and show some numerical examples. These are joint works with R. Carles (CNRS), L. Martaud and G. Beck (INRIA).
Gregory Falkovich
Weizmann Institute of ScienceOne Equation to Rule Them All, Weak and Strong
First, I will briefly review the weak turbulence theory from the perspective of theoretical physics. Second, an even shorter synopsis of quantum field theory will be given with the emphasis on renormalization of system parameters in a state with multi-scale fluctuations. Third and final, I shall describe an emerging theory applying the renormalization approach to wave turbulence. In particular, that will let us derive a closed (integro-differential) equation, which is able to describe wide classes of wave turbulence at any level of nonlinearity, from weak to strong. This so-called renormalized kinetic equation is a new interesting mathematical object. At a physical level, we shall see an emerging classification of the universality classes of wave turbulence, based on whether nonlinear interaction is enhanced or suppressed in the strong-turbulence regime.
Nader Masmoudi
New York UniversityRecent Advances in Nonlinear Inviscid Damping
Inviscid damping refers to the long-time decay of velocity perturbations in an ideal fluid, even though there’s no viscosity to dissipate the energy. This phenomenon is similar to the Landau damping in plasma physics. We review some old results and give some more recent advances about nonlinear inviscid damping. In particular, we will discuss the extension of the original result to more general shear flows. We will also discuss the optimality of the regularity spaces involved in some results by showing instability constructions. Joint results with J. Bedrossian, Yu Deng, Weiren Zhao.
Sylvia Serfaty
Sorbonne Université and Courant InstituteMean-Field Limits by the Modulated Energy Method
We will discuss joint works with Matt Rosenzweig, Antonin Chodron de Courcel, Hung Q. Nguyen, and Elias Hess-Childs, in which we study the question of mean-field limits, or deriving effective evolution equations of PDE type for a system of N points in singular interaction, for instance of Coulomb or Riesz, evolving by gradient flow or conservative flow (such as the point vortex system in 2D), with or without noise.
The convergence to the mean-field limit by the modulated energy method relies on a functional inequality of commutator estimate type. We will discuss various ways of proving the commutator estimate, and recent progress that provides sharp and localizable estimates.
We also discuss the question of obtaining global-in-time convergence and its connection with modulated log-Sobolev inequalities.
Gigliola Staffilani
Massachusetts Institute of TechnologySome Mathematically Rigorous Results in Wave Turbulence Theory
In this talk, we give an overview of some old and new mathematical advances in wave turbulence theory. We start with the original perspective of Bourgain on the study of the energy spectrum for a periodic nonlinear Schrödinger equation via growth of Sobolev norms. Then we move to the wave kinetic equations as effective equations for the energy spectrum, and we give examples of rigorous proofs of condensate growth and energy transfer. Some of the work presented is in collaboration with N. Camps and B. M. Tran.
Stephanie Waterman
University of British ColumbiaFilling in the Map: Understanding Arctic Ocean Mixing Rates, Mechanisms, Space-Time Variability, and Decadal-Scale Trends from Ocean Observations
The rates and mechanisms of ocean mixing are important controls on how the oceans function; yet, our understanding of mixing in the ocean is significantly limited by complex variability in mixing rates and processes, and by a scarcity of direct observations. In the Arctic Ocean, the challenges of understanding ocean mixing are significant: mixing measurements are especially sparse, and latitude, ice, and stratification make the ocean mixing environment unique.
In this talk, I’ll discuss our group’s work that uses a variety of ocean observations and observational methods to better understand Arctic Ocean mixing rates, mechanisms, space-time variability, and decadal-scale trends. Specifically, I will consolidate results from different studies to provide statistical characterizations of Arctic Ocean mixing rate and mixing flux distributions in time and space over a range of scales, as well as insights into the mechanisms driving and/or modulating the observed variability. I’ll argue that mixing in the Arctic Ocean spans multiple regimes with several important consequences: care is needed when inferring mixing rates using standard methods, the mixing of heat may be significantly enhanced above expectation, and the competition between stabilizing and destabilizing regimes may be an important control on stratification and ultimately mixing regime prevalence itself.
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Participation in the meeting falls into the following four categories. An individual’s participation category is communicated via their letter of invitation.
The Simons Foundation will never ask for credit card information or require payment for registration to our events.
Group A – Speakers & Organizers
Individuals in Group A receive travel and hotel coordination within the following parameters:
Travel
Economy Class: For flights that are three hours or less to your destination, the maximum allowable class of service is Economy class.
Premium Economy Class: For flights where the total air travel time (excluding connection time) is more than three hours and less than seven hours per segment to your destination, the maximum allowable class of service is premium economy.
Business Class: When traveling internationally (or to Hawaii/Alaska) travelers are permitted to travel in Business Class on those segments that are seven hours or more. If the routing is over budget, a premium economy or mixed-class ticket will be booked.Hotel
Up to 3 nights at the conference hotel, arriving on Wednesday, December 3, 2025 and departing on Saturday, December 6, 2025.Group B – Funded Participants
Individuals in Group B receive travel and hotel coordination within the following parameters:
Travel
Economy class travel will be booked regardless of flight length.Hotel
Up to 3 nights at the conference hotel, arriving on Wednesday, December 3, 2025 and departing on Saturday, December 6, 2025.Group C – Unfunded Participants
Individuals in Group C 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.
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Air and Rail
For funded individuals, the foundation will arrange and pay for round-trip travel from their home city to the conference city. All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency.
Travel Deviations
The following travel specifications are considered deviations and will only be accommodated if the cost is less than or equal to the amount the Simons Foundation would pay for a standard round-trip ticket from your home city to the conference city:
- Preferred airline
- Preferred travel class
- Specific flights/flight times
- Travel dates outside those associated with the conference
- Arriving or departing from an airport other than your home city or conference city airports, i.e. multi-segment or triangle trips.
All deviations must be reviewed and approved by the Simons Foundation and, if the cost is more than what would normally be paid, a reimbursement quote must be obtained through the foundation’s travel agency before proceeding to booking and paying for travel out of pocket. All reimbursements for travel booked directly will be paid after the conclusion of the meeting.
Changes After Ticketing
All costs related to changes made to ticketed travel are to be paid for by the participant and are not reimbursable. Please contact the foundation’s travel agency for further assistance.
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 Hotel Seville NoMad 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.
Hotel
Funded individuals who require hotel accommodations are hosted by the foundation for a maximum of 3 nights at the conference hotel, arriving on Wednesday, December 3, 2025 and departing on Saturday, December 6, 2025.
Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link included in your invitation.
Hotel Seville NoMad
22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)
https://www.hyatt.com/unbound-collection/en-US/nycud-hotel-seville-nomadFor driving directions to the Hotel Seville NoMad, please click here.
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Overview
In-person participants 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 because of meetings not directly related to the Simons Foundation-hosted meeting (a satellite meeting or 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.
Meals
The daily meal limit is $125; itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses up the following amounts.
- Breakfast $20
- Lunch $30
- Dinner $75
Allowable Meal Expenses
- Meals taken on travel days (when you traveled by air or train).
- Meals not provided on a meeting day, dinner on Friday for example.
- Group dinners consisting of fellow meeting participants paid by a single person will be reimbursed up to $75 per person and the amount will count towards the $125 daily meal limit.
Unallowable Meal Expenses
- Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner).
- Meals taken on days not associated with Simons Foundation-coordinated events.
- Minibar expenses.
- Meal expenses for a non-foundation guest.
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Ubers, Lyfts, taxis, etc., taken to and from restaurants in Manhattan.
- Accommodations will be made for those with mobility restrictions.
Ground Transportation
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport or train station) as well as subway and bus fares while in Manhattan are reimbursable.
Transportation to/from satellite meetings are not reimbursable.
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Attendance
In-person participants and speakers are expected to attend all meeting days. Participants receiving hotel and travel support wishing to arrive on meeting days which conclude at 2:00 PM will be asked to attend remotely.
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.
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Meeting & Policy Questions
Christina Darras
Event Manager
[email protected]Travel & Hotel Support
FCM Travel Meetings & Events
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