Mean Field Games: Well-posedness, Singularities and Beyond (?)

• Pierre-Louis Lions (Collège de France)

Room: Centre de conférences Marilyn et James Simons

The Small Data Global Well-posedness Conjecture for 1D Defocusing Dispersive Flows

• Daniel Tataru (UC Berkeley)

Room: Centre de conférences Marilyn et James Simons The conjecture broadly asserts that small data should yield global solutions for 1D defocusing dispersive flows with cubic nonlinearities, in both semilinear and quasilinear settings. The aim of the talk will be to present some very recent results in this direction. This is joint work with Mihaela Ifrim.

The Quartic Integrability and Long Time Existence of Steep Water Waves in 2d

• Sijue Wu (University of Michigan)

Room: Centre de conférences Marilyn et James Simons

The Nonlinear Stability of Slowly Rotating Kerr Black Holes

• Sergiu Klainerman (Princeton University)

Room: Centre de conférences Marilyn et James Simons I will talk about my joint work with J. Szeftel, and partly also with E. Giorgi and D. Shen, on the nonlinear stability of slowly rotating Kerr black holes.

Area-minimizing Integral Currents: Singularities and Structure

• Camillo De Lellis (IAS)

Room: Centre de conférences Marilyn et James Simons Area-minimizing integral currents are a natural generalization of area-minimizing oriented surfaces. The concept was pioneered by De Giorgi for hypersurfaces of the Euclidean space, and extended by Federer and Fleming to any codimension and general Riemannian ambients. Celebrated examples of singular $7$-dimensional minimizers in $\mathbb{R}^8$ and of singular $2$-dimensional minimizers in $\mathbb{R}^4$ are known since long. Moreover, a theorem which summarizes the work of several mathematicians in the 60es and 70es (De Giorgi, Fleming, Almgren, Simons, and Federer) and a celebrated work by Almgren in the 80es give dimension bounds for the singular set which match the one of the examples, in codimension 1 and in general codimension respectively. In codimension, higher than 1 a recent result of Liu shows that the singular set can in fact be a fractal of any Hausdorff dimension $\alpha \leq m -2$. On the other hand, it seems likely that it is an $(m -2)$-rectifiable set, i.e. that it can be covered by countably many $C^1$ submanifolds leaving aside a set of zero $m-2$-dimensional Hausdorff measure. This conjecture is the counterpart of a celebrated work of Leon Simon in the nineties for the hypersurface case. In this talk I will report on progress towards its proof, based on recent joint works with Anna Skorobogatova and Paul Minter.

Instability and Non-uniqueness for the Euler and Navier-Stokes Equations

• Maria Colombo (EPFL)

Room: Centre de conférences Marilyn et James Simons The incompressible Navier-Stokes and Euler equations are fundamental PDEs in mathematical fluid dynamics and their well-posedness theory is nowadays largely open. The past decade has seen a surprising and remarkable progress, through various different attempts, in describing some non-unique solutions of these PDEs. The talk will survey some of the recent contributions in this direction, including works in collaboration with Albritton and Brué which show that Leray-Hopf solutions of the forced Navier-Stokes equations are not unique.

Waves, Disorder and Interactions: a Physicist's Perspective

• Thierry Giamarchi (Université de Genève)

Room: Centre de conférences Marilyn et James Simons As discovered in the seminal paper of P. W. Anderson in 1958 when an equation such as the Schroedinger 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. This phenomenon, the so-called Anderson localization indicates that a single quantum particle in such random landscape would be localized. 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. I will give in this talk an overview of this class of phenomena and point towards some of the challenges in the field. 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

The Stability-compactness Method and Qualitative Properties of Nonlinear Elliptic Equations

• Henri Berestycki (EHESS)

Room: Centre de conférences Marilyn et James Simons In this talk, I report on a series of works with Cole Graham on semi-linear elliptic equations with positive non-linearities. Solutions represent stationary states of reaction-diffusion equations. We focus on qualitative properties such as uniqueness, symmetries, and stability. The main motivation is to study these equations in general unbounded domains, which exhibit remarkably rich behavior. Our method rests on decomposing the problem into a compact part and one for which a stability result can be derived and then to combine the two. This approach has proved to be unexpectedly versatile and in fact, encompasses past works on the subject such as the general moving plane method.

Stability and Asymptotic Behavior of Fronts in Bidomain Models

• Hiroshi Matano (Meiji University)

Room: Centre de conférences Marilyn et James Simons

Global Behavior of 3D Shocks and Landau Law of Decay

• Igor Rodnianski (Princeton University)

Singular Stochastic PDE: More Geometry and Less Combinatorics

• Felix Otto (Max-Planck Institut fur Mathematik)

Singular stochastic PDE are those stochastic PDE in which the noise is so rough that the nonlinearity requires a renormalization. The guiding principle of renormalization is to preserve as many symmetries of the solution manifold as possible. We follow the typical approach of mathematical physics, and of Hairer’s regularity structures, which provides a formal series expansion of a general solution. However, we advocate a more geometric/analytic than combinatorial version of this approach: Instead of appealing to an expansion in- indexed by trees, we consider all partial derivatives w. r. t. the “constitutive” function defining the nonlinearity. Instead of a Gaussian calculus guided by Feynman diagrams arising from pairing nodes of two trees, we consider derivatives w. r. t. the noise, i.e. Malliavin derivatives. This calculus allows to characterization the expansion without divergent terms; in conjunction with the spectral gap estimate, it provides a natural path toward stochastic estimates. This is joint work with P. Linares, M. Tempelmayr, and P. Tsatsoulis, based on work with J. Sauer, S. Smith, and H. Weber.

Solitons and Channels

• Carlos Kenig (University of Chicago)

We will discuss the role of non-radiative solutions to nonlinear wave equations, in connection with soliton resolution. Using new channels of energy estimates we characterize all radial non-radiative solutions for a general class of nonlinear wave equations. This is joint work with C.Collot, T. Duyckaerts and F. Merle.

Probabilistic and Deterministic Scattering for Non-linear Schrödinger Equations

• Nicolas Burq (Université Paris-Saclay)

In this talk, I will present results on the scattering for non-linear Schrödinger equations with random initial data. I will also show how some ideas from this probabilistic perspective lead to new results in the description of the wave operators for deterministic scattering. This is based on joint works with H. Koch, N. Visciglia, and N. Tzvetkov

Blow up for the 1d Cubic NLS and Related Systems

• Luis Vega (BCAM)

We will review some recent results about the formation of singularities for the one-dimensional Schrödinger equation with cubic nonlinearity. The connection will be also established with the Schrödinger map and the vortex filament equation.

Mean-Field Limits for Singular Flows

• Sylvia Serfaty (New-York University)

Room: Centre de conférences Marilyn et James Simons We consider a system of N points in singular interaction of Coulomb or Riesz type, evolving by gradient flow or conservative flow (such as the point vortex system in 2D) with or without noise. We discuss convergence to the mean-field limit by a modulated energy method, that relies on a commutator estimate. The method also allows to obtain global-in-time convergence

The Regularity Problem for the Landau Equation

• François Golse (École polytechnique)

Room: Centre de conférences Marilyn et James Simons

Invariant Gibbs Measures for 2D NLS and 3D Cubic NLW

• Andrea Nahmod (University of Massachusetts)

Room: Centre de conférences Marilyn et James Simons In this talk, we discuss recent developments in the study of the propagation of randomness under the flow of dispersive PDE. In particular, we present a non-technical overview of recent works that led to the resolution of two open problems concerning Gibbs measure invariance for the 2D NLS with arbitrary wave interactions (joint with Yu Deng and Haitian Yue) and for the 3D cubic NLW (joint with Bjoern Bringmann, Yu Deng and Haitian Yue). The first one is proved using the method of random averaging operators, while the second one relies on the theory of random tensors in conjunction with other techniques, such as paracontrolled calculus and heat-wave analysis.

Reversal in the Stationary Prandtl Equations

• Nader Masmoudi (New-York University)

Room: Centre de conférences Marilyn et James Simons We investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by regions in which $u > 0$ and $u < 0$ respectively. The classical point of view of regarding the stationary Prandtl system as an evolution equation in $x$ completely breaks down since $u$ changes sign. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. This is a joint work with Sameer Iyer.

35 Years of Critical Nonlinear Analysis

• Pierre Raphaël (University of Cambridge)

Room: Centre de conférences Marilyn et James Simons

Dispersive Estimates for the Semi classical Schrödinger Flow inside General Convex Domains and Applications to the Cubic NLS in 3D

• Oana Ivanovici (CNRS, Sorbonne Université)

Room: Centre de conférences Marilyn et James Simons We obtain fixed time decay rate for the linear semi-classical Schrödinger flow inside a general strictly convex domain. Corresponding Strichartz estimates allow to solve the cubic NLS on such 3D convex domains. This is joint work with F.Planchon

Singularity Models in 3D Ricci Flow

• Simon Brendle (Columbia University)

Room: Centre de conférences Marilyn et James Simons The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. From a PDE perspective, the Ricci flow is a system of linear parabolic equations, which can be viewed as the heat equation analog of the Einstein equations in general relativity. The central problem in the field is to understand singularity formation. In other words, what does the geometry look like at points where the curvature is large? In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension 3; this is sufficient for topological conclusions. In this lecture, we will discuss recent developments which have led to a complete classification of all the singularity models in dimension 3.

On Continuous Time Bubbling for the Harmonic Map Heat Flow in Two Dimensions

• Wilhelm Schlag (Yale University)

Room: Centre de conférences Marilyn et James Simons I will describe recent work with Jacek Jendrej (CNRS, Paris Nord) and Andrew Lawrie (MIT) on harmonic maps of finite energy from the plane to the two sphere, without making any symmetry assumptions. While it has been known since the 1990s that bubbling occurs along a carefully chosen sequence of times via an elliptic Palais-Smale mechanism, we show that this continues to hold continuously in time. The key notion is that of the “minimal collision energy” which appears in the soliton resolution result by Jendrej and Lawrie on critical equivariant wave maps.