Open Issues in Gravitation

• Thibault Damour (IHES)

Room: Centre de conférences Marilyn et James Simons After briefly reviewing some of the high-precision experimental confirmations of General Relativity (GR), the talk will discuss some open issues concerning both physical and mathematical aspects of GR, notably: . Tantalizing links between GR and Yang-Mills theory ; . Links between Einstein equations and Navier-Stokes equations; . Emergence of hidden symmetry structures near cosmological singularities; . Peeling properties and symmetry structures near infinity; . Skeletonization of Black Holes as point masses; . Scarcity of viable extensions of GR.

Two Methods for Deriving Singular Mean-Field Limits

• Sylvia Serfaty (Sorbonne Université)

Room: Centre de conférences Marilyn et James Simons We are interested in 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 nature, evolving by first order dynamics. We will discuss two methods: the modulated energy method, that works well for gradient flows or conservative flows of Coulomb/Riesz type energies, and a new method based on a multiscale mollification metric, which works well for up to Coulomb interaction singularity, without much structure assumed.

Furstenberg Sets Estimates with Application to Restriction Theory

• Hong Wang (IHES & NYU)

Room: Centre de conférences Marilyn et James Simons A Kakeya set is a compact subset of $\mathbb{R}^n$ containing a unit line segment in every direction. More generally, for $0 < s \leq 1$, an $s$-Furstenberg set is a subset $E \subset \mathbb{R}^n$ such that for every direction there is a unit line segment whose intersection with $E$ has Hausdorff dimension at least $s$. Furstenberg set problems ask for lower bounds on ${\rm dim}_H(E)$ in terms of $s$ and $n$. In this talk I will discuss how such dimension estimates arise naturally in Fourier restriction theory via wave packet decompositions. From this perspective it is natural to consider s-dimensional subsets of line segments, rather than whole segments, because waves may concentrate on sparser subsets of tubes. This is based on joint work with Shukun Wu and joint work in progress with Dima Zakharov.

The Role of Spacetime Geometry in Gas Dynamics

• Pin Yu (Tsinghua University)

Room: Centre de conférences Marilyn et James Simons In the first part, we examine how the geometric framework of the underlying spacetime provides powerful insights for analyzing wave equations, with a focus on the pioneering ideas of S. Klainerman in the 1980s and his collaborative work with D. Christodoulou on the nonlinear stability of Minkowski spacetime. The second part of the talk will demonstrate applications to the compressible Euler equations, revealing critical perspectives into shock formation mechanisms, the construction of centered rarefaction waves, and the structure of the singularities.

The Einstein Equation in Kähler Geometry

• Duong Phong (Columbia University)

Room: Centre de conférences Marilyn et James Simons In Kähler geometry, the Einstein equation reduces to a scalar equation. The existence of a solution to this equation was conjectured by E. Calabi in the 1950’s and subsequently proved by S.T. Yau in the mid 1970’s. But recent advances building on Yau’s theorem can go much further, and accumulate a wealth of geometric information for Kähler manifolds, including diameter and non-collapse volume estimates, Green’s functions, Sobolev inequalities, and improved versions of the Gromov convergence theorem, none of which requires any assumption on the Ricci curvature, as their Riemannian analogues do. This is joint work with B. Guo, F. Tong, J. Song, and J. Sturm.

On the Wave Turbulence Theory of 2D Gravity Water Waves

• Alexandru Ionescu (Princeton University)

Room: Centre de conférences Marilyn et James Simons I will talk about some recent work on the problem of establishing rigorously a wave turbulence theory for water waves systems. This is a classical problem in Mathematical Physics, going back to pioneering work of Hasselmann. To address it we propose a new mechanism, based on a combination of two main ingredients: (1) deterministic energy estimates for all solutions that are small in $L^\infty$-based norms, and (2) probabilistic arguments aimed at understanding propagation of randomness on long time intervals. This is joint work with Yu Deng and Fabio Pusateri.

Anomalous Diffusivity and Regularity for Random Incompressible Flows

• Scott Armstrong (Sorbonne Université)

Room: Centre de conférences Marilyn et James Simons I will present work on the long-time behavior of Brownian motion in a stationary, incompressible random drift field with slowly decaying correlations. In this setting one expects the variance of the displacement to grow faster than linearly in time, with an exponent determined by the correlation structure of the drift (as predicted by Bouchaud-Georges in 1990). We view the problem through the associated divergence-form drift-diffusion operator and apply a scale-by-scale coarse-graining scheme to its coefficients. This produces, at each scale, an effective Laplacian whose diffusivity depends on the scale, together with quantitative control of the error of this approximation. This can be seen as a rigorous version of the perturbative renormalization group heuristics proposed by Bouchaud-Georges. A crucial role is played by anomalous regularization, that is, elliptic estimates that are independent of the bare molecular diffusivity. This work I describe is joint with A. Bou-Rabee and T. Kuusi.

A Proof of Onsager’s Conjecture for the Incompressible Euler Equations

• Philip Isett (Caltech)

Room: Centre de conférences Marilyn et James Simons In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-$\epsilon$)-Hölder Euler flows in 3D that have compact support in time.

Late-time Tails of Linear and Nonlinear Waves

• Jonathan Luk (Stanford University)

Room: Centre de conférences Marilyn et James Simons I will present a joint work with Sung-Jin Oh (Berkeley), where we develop a general method for understanding the precise late-time asymptotic behavior of solutions to linear and nonlinear wave equations in odd spatial dimensions. In particular, we prove that in the presence of a nonlinearity and/or a dynamical background, the late time tails are in general different from the better understood case of linear equations on stationary backgrounds. I will explain how the late time tails are related to the problem of the singularity structure in the interior of generic dynamical vacuum black holes in general relativity.

The Nonlinear Stability of Kerr and Kerr–Newman Black Holes

• Elena Giorgi (Columbia University)

Room: Centre de conférences Marilyn et James Simons Kerr and Kerr–Newman black holes are asymptotically flat stationary solutions of the Einstein and Einstein–Maxwell equations exhibiting horizons, trapping, and subtle dispersive dynamics. I will survey recent progress on their nonlinear stability, focusing on the proof of the full nonlinear stability of the slowly rotating Kerr family as solutions of the vacuum Einstein equations, obtained in a series of works by Klainerman, Szeftel, Shen and myself, and briefly discuss how the presence of an electromagnetic field and charge modifies the dynamics in the Kerr–Newman setting. These results build on the pioneering contributions of Sergiu Klainerman to the analysis of nonlinear wave equations and the global dynamics of the Einstein equations.

A Physical-space Approach to Global Asymptotics for Variable-coefficient Klein-Gordon Equations

• Sung-Jin Oh (UC Berkeley)

Room: Centre de conférences Marilyn et James Simons In this talk, I will discuss a new physical-space approach to establish the time decay and global asymptotics of solutions to variable-coefficient Klein-Gordon equations in (3+1)-dimensions. A key innovation in our methodology is the concept of a “good commutator,” which extends Klainerman’s classical commuting vector field method, and which combines well with Ifrim-Tataru’s testing by wave packets. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Klein-Gordon equations with quadratic nonlinearity, variable coefficients in their linear part, and possibly outside obstacles. This talk is based on an upcoming work with F. Pasqualotto (UCSD) and N. Tang (UC Berkeley).

The Soliton Resolution Conjecture for the Benjamin-Ono Equation

• Patrick Gérard (Univ. Paris-Saclay, LMO)

Room: Centre de conférences Marilyn et James Simons The soliton resolution conjecture for a dispersive equation claims that, in the long time asymptotics, every solution decouples as a sum of soliton solutions up to a radiative remainder. I will discuss a recent proof of this conjecture in the special case of the Benjamin-Ono equation ( in collaboration with Louise Gassot and Peter Miller).

On Global Dynamics of 3-D Irrotational Compressible Fluids

• Qian Wang (University of Oxford)

Room: Centre de conférences Marilyn et James Simons We consider global-in-time evolution of irrotational, isentropic, compressible Euler flow in 3-D. We study a broad class of smooth Cauchy data, prescribed on an annulus and surrounded by a non-vacuum constant exterior state, without symmetry assumptions. By imposing a sufficient expansion condition on the initial data and using the nonlinear structure of the Euler equations, we show that the first-order transversal derivative of the normalized density decays as ⟨t⟩⁻¹ (log⟨t⟩ + 1)⁻¹, provided that the perturbation arising from the tangential derivatives can be properly controlled for all t by using a bootstrap argument. This enables us to construct global exterior solutions, including a rather general subclass forming rarefaction at null infinity. Our result applies to data with a total energy of any size, as it does not require smallness of the transversal derivatives of smooth data.

Minimal Surfaces Defined by Extremal Eigenvalue Problems

• Richard M. Schoen (UC Irvine)

Room: Centre de conférences Marilyn et James Simons Minimal surfaces in spheres are characterized by the condition that their embedding functions are eigenfunctions on the surface with its induced metric. The metric on the surface turns out to be an extremal for the eigenvalue among metrics on the surface with the same area. In recent decades, this extremal propertyhas been used to construct new minimal surfaces by eigenvalue maximization. There is an analogous theory for minimal surfaces in the euclidean ball with a free boundary condition. In this talk we will describe new work that generalizes this idea to products of balls. We will describe the general theory and apply it in a specific case to explain and generalize the Schwarz p-surface, which is a free boundary minimal surface in the three dimensional cube with one boundary component on each face of the cube. We will show how the method can be used to construct such surfaces in rectangular prisms with arbitrary side lengths.

On Stable Implosions

• Vlad Vicol (NYU)

Room: Centre de conférences Marilyn et James Simons We exhibit a new class of self-similar implosion solutions for the full compressible Euler equations. For any value of the adiabatic exponent, we construct a sequence of implosion profiles that are smooth before collapse and have an explicit similarity exponent. The first profile in this sequence (the ''ground state'') possesses remarkable stability properties, even outside of spherical symmetry. This is joint work with J. Chen (U Chicago) and S. Shkoller (UC Davis).

Rogue Waves and Large Deviations for the 2D Pure Gravity Deep Water Wave Problem

• Gigliola Staffilani (MIT)

Room: Centre de conférences Marilyn et James Simons Rogue waves are extreme ocean events characterized by the sudden formation of anomalously large crests. This phenomenon remains an important subject of investigation in oceanography and mathematics. A central problem is to quantify the probability of their formation under random Gaussian sea initial data. We rigorously justify the exponential large-tail law probability for the formation of rogue waves of the pure gravity water wave equations in deep water, up to the optimal timescales allowed by deterministic well-posedness theory. The proof shows that rogue waves most likely arise through “dispersive focusing”, where phase synchronization produces constructive amplification of the water crest. The main difficulty in justifying this mechanism is propagating statistical information over such long timescales, which we overcome by combining normal forms and probabilistic methods. Unlike previous results, this novel approach does not require approximate solutions to be Gaussian. This is joint work with M. Berti, R. Grande and A. Maspero

Waves, Nonlinearity and Geometry or How Sergiu Klainerman Has Influenced Generations of Mathematicians

• Jacques Smulevici (Sorbonne Université)

Room: Centre de conférences Marilyn et James Simons Over the past four decades, the analysis of nonlinear wave equations has changed dramatically, with the emergence of many new methods that have led to remarkable and influential results. Central to this transformation is the work of Sergiu Klainerman, which has reshaped the analysis of hyperbolic equations and influenced several generations of researchers. This talk will survey some of Klainerman’s seminal ideas and results, beginning with his earlier work on the analysis quasilinear wave equations in the large, the commuting vector field method and the null condition. I will then describe how this framework extended naturally into geometric settings, leading to major breakthroughs such as the nonlinear stability of Minkowski space. I will also discuss how related ideas and techniques influenced other fundamental developments in mathematical relativity, including results on the uniqueness and rigidity of black holes, the proof of the $L^2$ curvature theorem and the more recent developments on the stability of black holes. Throughout the talk, I will reflect on how these ideas have influenced the works of many other mathematicians, including work I have been involved in, and continue to shape the field today.

Long Time and Global Dynamics in Nonlinear Dispersive Flows

• Daniel Tataru (UC Berkeley)

Room: Centre de conférences Marilyn et James Simons The key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. Nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand. The first goal of this talk will be to describe a recent set of conjectures which aim to characterize the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. This covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. The second objective of the talk will be to describe some very recent results in this direction, in joint work with Mihaela Ifrim.

Boundary Plateau Laws

• Camillo De Lellis (IAS)

Room: Centre de conférences Marilyn et James Simons Dipping a wire of metal or plastic in soapy water and taking it out is a favorite classroom experiment: typically the soapy water will form a thin film which is attached to the wire. The classical Plateau laws, stated by the Belgian physicist Joseph Plateau in the nineteenth century, assert that, away from the wire, the local geometry of a soap film is described locally by the following list of shapes: a 2-dimensional plane, three halfplanes meeting at a common line with equal angles, and the cone over the $1$-dimensional skeleton of a regular tetrahedron. Is there a similar list of possible shapes for the points where the film touches its ``boundary'', namely the wire of the classroom experiment? The classical Plateau laws were translated into a mathematical theorem by Jean Taylor in the seventies: in a nutshell Taylor's theorem rigorously classifies 2-dimensional conical shapes which minimize the area. In this talk I will illustrate a recent joint work with Federico Glaudo, classifying conical shapes which minimize the area and include a boundary line: the corresponding list suggests an analog of Plateau's laws at the boundary of the soap film, which are very much in agreement with both real-life and numerical experiments.