Oct 19, 2015
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Wave turbulence for the cubic Szegö equation and beyond
Patrick Gerard (Université de Paris XI)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This talk will be divided into two parts.
The first part will be devoted to the introduction of the cubic Szegö equation on the circle and of its integrable structure, with applications to existence of solutions displaying wave turbulence.
The second part will give an idea of works in progress about extensions of wave turbulence to related Hamiltonian PDEs.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Diffusive limits for stochastic kinetic equtions
Arnaud Debussche (Ecole Normale Supérieure de Rennes)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk, we consider kinetic equations containing random
terms. The kinetic models contain a small parameter and it is well
known that, after scaling, when this parameter goes to zero the limit
problem is a diffusion equation in the PDE sense, ie a parabolic equation
of second order. A smooth noise is added, accounting for external perturbation.
It scales also with the small parameter. It is expected that the limit
equation is then a stochastic parabolic equation where the noise is in
Stratonovitch form.
Our aim is to justify in this way several SPDEs commonly used.
We first treat linear equations with multiplicative noise. Then show how
to extend the methods to nonlinear equations or to the more physical
case of a random forcing term.
The results have been obtained jointly with S. De Moor and J. Vovelle
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Symplectic nonsqueezing for the cubic NLS on R^2
Monica Visan (University of California, Los Angeles)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We prove that the flow of the masscritical NLS in two dimensions cannot squeeze a ball in $L^2$ into a cylinder of lesser radius. This is a PDE analogue of Gromov's nonsqueezing theorem for an infinitedimensional Hamiltonian PDE in infinite volume. This is joint work with R. Killip and X. Zhang
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Optimal Regularity for some Parabolic SPDEs
Davar Khoshnevisan (University of Utah)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
. I will present some recent optimal regularity results for parabolic SPDEs driven by spacetime white noise.
Our results are connected closely to questions about how sensitive the solution to a parabolic SPDE is to small changes in the initial data.
This is based on joint work with Le Chen and Kunwoo Kim
 Supplements




Oct 20, 2015
Tuesday

09:30 AM  10:30 AM


An Applied Math Perspective on Climate Science, Turbulence, and Other Complex Systems
Andrew Majda (New York University, Courant Institute)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Generalized Smoluchowski Equations and Scalar Conservation Laws
Fraydoun Rezakhanlou (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
By a classical result of Bertoin, if initially a solution to Burgers' equation is a Levy process without positive jumps, then this property persists at later times. According to a theorem of Groeneboom, a white noise initial data also leads to a Levy process at positive times. Menon and Srinivasan observed that in both aforementioned results the evolving Levy measure satisfies a Smoluchowski–type equation. They also conjectured that a similar phenomenon would occur if instead of Burgers' equation, we solve a general scalar conservation law with a convex flux function. Though a Levy process may evolve to a Markov process that in most cases is not Levy. The corresponding jump kernel would satisfy a generalized Smoluchowski equation. Along with Dave Kaspar, we show that a variant of this conjecture is true for monotone solutions to scalar conservation laws. I also formulate some open question concerning the analogous questions for HamiltonJacobi PDEs in higher dimensions
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


The Talbot effect and the evolution of Vortex Filaments
Luis Vega (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I shall present some recent work done in collaboration with F. De la Hoz about the the evolution of regular polygons within the socalled Vortex Filament Equation. Each corner of the polygon generates some Kelvin waves that interact in a nonlinear way that is closely related to the (linear) Talbot effect in optics.
The question of the possible connection between the Talbot effect and turbulence will be also addressed, and in particular the appearance of multifractals and their relation with the so called FrischParisi conjecture.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


KPII in 2 and 3d
Herbert Koch (Rheinische FriedrichWilhelmsUniversität Bonn)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In the talk I will explain global wellposedness and scattering results for dispersive equations with quadratic nonlinearities. The KadomtsevPetviashvili II equation in 3d requires additional decompositions, related function spaces and refined bilinear estimates.
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Oct 21, 2015
Wednesday

09:30 AM  10:30 AM


The KardarParisiZhang equation and universality class
Jeremy Quastel (University of Toronto)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We survey recent progress on asymptotic fluctuations in the KPZ universality class
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


On twodimensional gravity water waves with angled crests
Sijue Wu (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk, I will present our recent work on the local in time existence of twodimensional gravity water waves with angled crests. Specifically, we construct an energy functional $E(t)$ that allows for angled crests in the interface. We show that for any initial data satisfying $E(0)<\infty$, there is $T>0$, depending only on $E(0)$, such that the water wave system is solvable for time $t\in [0, T]$. Furthermore we show that for any smooth initial data, the unique solution of the 2d water wave system remains smooth so long as $E(t)$ remains finite.
 Supplements



Oct 22, 2015
Thursday

09:30 AM  10:30 AM


Exotic blow up rates for some critical nonlinear dispersive equations
Yvan Martel (École Polytechnique)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will review some recent results on the construction of blow up solutions with exotic blow up rate for some critical nonlinear dispersive equations
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Toward a smooth ergodic theory for infinite dimensional systems
LaiSang Young (New York University, Courant Institute)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Focusing on settings that are consistent with semiflows defined by
dissipative parabolic PDEs, I will discuss some first steps toward
building a dynamical systems theory, in particular a theory of chaotic
systems, for maps and semiflows in Hilbert and Banach spaces.
I will survey known results and present recent progress, including
theorems on Lyapunov exponents, periodic solutions and horseshoes,
entropy formula and SRB measures, and a notion of “almost every”
initial condition that is natural to the underlying dynamics. Technical
differences between finite and infinite dimensions will also be discussed
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Invariant measures and the soliton resolution conjecture
Sourav Chatterjee (Stanford University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will talk about the microcanonical invariant measure for the discrete nonlinear Schrödinger equation on a torus in the masssubcritical regime, and prove that a random function drawn from this measure is close to the ground state soliton with high probability. This proves that “almost all” ergodic components of this flow have the property of convergence to a soliton in the long run, which is a statistical variant of what is sometimes called the soliton resolution conjecture
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


The large box limit of nonlinear Schrodinger equations in weakly nonlinear regime
Jalal Shatah (New York University, Courant Institute)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We study the long time dynamics of solutions to nonlinear Schrodinger equations with periodic boundary conditions as the length of the period becomes infinite. We isolate the effects of resonant interactions and derive new evolution equations whose dynamics approximate the long time dynamics of localized solutions. We will show that this approximation is valid on a long time scale determined by the size of the solution and the length of the period.
 Supplements



Oct 23, 2015
Friday

09:30 AM  10:30 AM


Motion of a Random String
Martin Hairer (Imperial College, London)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Random Versus Deterministic Approach in the Study of Wave and Dispersive Equations
Gigliola Staffilani (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The point of this talk is to show how certain wellposedness results that are not available using deterministic techniques involving Fourier and harmonic analysis
can be obtained when introducing randomization in the set of initial data. Along the way I will also prove a certain “probabilistic propagation of regularity” for certain almost sure globally wellposed dispersive equations. This talk is based on joint work with A. Nahmod
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Energy distribution and wave turbulence closures for the nonlinear Schrodinger equation
Zaher Hani (Georgia Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We introduce the theory of wave turbulence as a systematic approach to studying the energy distribution across scales in nonlinear dispersive PDE. This is done by deriving an effective equation for the energy density of the system in a statistical setting, by taking weak nonlinearity and infinite volume limits. The resulting equation is called the “wave kinetic equation”, and it gives, at a formal level, a lot of insight into the outofequilibrium dynamics and statistics of nonlinear dispersive systems. The fundamental problem here is to give a rigorous derivation of this formally derived equation. Without any stochastic element in the system, such problems are often too difficult to resolve (even in much simpler ODE settings). We will show how this equation can be derived starting from the nonlinear Schrodinger equation on a large torus, in the presence of an appropriate random force, and in the weakly nonlinear infinite volume limit. This is joint work with Isabelle Gallagher and Pierre Germain
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Classical Hamiltonian Systems, Driven out of Equilibrium, a Review
JeanPierre Eckmann (University of Geneva)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
I will try to explain, starting from the historical papers, what has been understood (and not understood) from a rigorous point of view, about classical Hamiltonian systems. For example a chain of massive bodies, or a coupled set of rotators, which are driven out of equilibrium by stochastic forces acting on the ends of the chain
 Supplements




Oct 26, 2015
Monday

09:30 AM  10:30 AM


From particles to linear hydrodynamic equations
Thierry Bodineau (École Polytechnique)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We consider a tagged particle in a diluted gas of hard spheres. Starting from the hamiltonian dynamics of particles in the BoltzmannGrad limit, we will show that the tagged particle follows a Brownian motion after an appropriate rescaling. We use the linear Boltzmann equation as an intermediate level of description for one tagged particle in a gas close to global equilibrium
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


From particles to linear hydrodynamic equations
Isabelle Gallagher (École Normale Supérieure)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We derive the linear acoustic and StokesFourier equations as the limiting dynamics of a system of hard spheres in a diluted gas in two space dimensions. We assume the system is initially close to equilibrium and we use the linearized Boltzmann equation as an intermediate step.
Joint work with T. Bodineau, L. SaintRaymond
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Global stability of a flat interface for the gravitycapillary waterwave model
Benoit Pausader (Brown University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The water wave model describes the evolution of a flat interface between air and an inviscid, incompressible fluid. It is known that (in 3D), if one considers the action of either gravity or surface tension alone, small localized perturbations of a flat interface lead to global solutions that scatter back to equilibrium, in a joint work with Y. Deng, A. Ionescu and F. Pusateri, we show that this remains true when one considers both forces.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


On global solutions of water wave models
Alexandru Ionescu (Princeton University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss some recent work, joint with Yu Deng, Benoit Pausader, and Fabio Pusateri, on the construction of global solutions of several water wave models. Our work concerns mainly the gravitycapillary model in 3D. I will also discuss the more general twofluid interface problem
 Supplements



Oct 27, 2015
Tuesday

09:30 AM  10:30 AM


Control of water waves
Thomas Alazard (Ecole Normale Supérieure ParisSaclay; Centre National de la Recherche Scientifique (CNRS))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Water waves are disturbances of the free surface of a liquid. They are, in general, produced
by the immersion of a solid body or by impulsive pressures applied on the free surface. The
two questions we discuss in this talk are the following:
 which waves can be generated by blowing on a localized portion of the free surface ?
Our main result with Pietro Baldi and Daniel HanKwan asserts that one can generate any small amplitude, periodic in x, twodimensional, gravitycapillary water waves.
 consider now gravity water waves in a rectangular tank. Isit possible to estimate their energy by
looking only at the motion of some of the curves of contact between the free surface and the vertical walls
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Universality in polytope phase transitions and message passing algorithms
Andrea Montanari (Stanford University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We consider a class of nonlinear mappings F
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


The stochastic LandauLifshitz equation
Anne de Bouard (École Polytechnique)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will review some recent results on the stochastic LandauLifshitz equation which models temperature effects on magnetization dynamics in micromagnetism. The particularity of the model is the geometric constraint the magnetization is a unit vector — which makes the equation nonlinear. The associated stochastic partial differential equation, taking account of temperature effects, has known some recent improvements, from the point of view of mathematical analysis as well as numerical analysis, but still offers a lot of open problems
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


On long term dynamics of nonlinear evolution equations
Wilhelm Schlag (University of Chicago)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will review some of the work conducted over the past five years
on the asymptotic behavior of solutions to dispersive nonlinear evolution equations
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Oct 28, 2015
Wednesday

09:30 AM  10:30 AM


Dispersion for the wave and the Schrödinger equations outside strictly convex domains
Oana Ivanovici (Sorbonne University)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
We consider a (general) strictly convex domain in R^d of dimension d>1 and we describe dispersion for both wave and Schrödinger equations with Dirichlet boundary condition. If d=2 or d=3 we show that dispersion does hold like in the flat case, while for d>3, we show that there exist strictly convex obstacles for which a loss occur with respect to the boundary less case (such an optimal loss is obtained by explicit computations).
 Supplements



10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


SPDEs on graphs as limit of SPDEs on narrow channels
Sandra Cerrai (University of Maryland)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 Supplements



Oct 29, 2015
Thursday

09:30 AM  10:30 AM


Renormalisation in regularity structures
Lorenzo Zambotti (Université de Paris VI (Pierre et Marie Curie))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We describe a general approach to the renormalisation step in the theory of regularity structures based on Hopf algebras of labelled trees (joint work with Yvain Bruned and Martin Hairer)."
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Blowup of the critical norm for supercritical wave equations
Thomas Duyckaerts (Université de Paris XIII (ParisNord))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk (essentially based in a joint work with Tristan Roy), I will review some recent results about the blowup of the scaleinvariant Sobolev norm for nonlinear wave equations in space dimension 3.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Second microlocalization and stabilization of damped wave equations on tori
Nicolas Burq (Université de Paris XI)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We consider the question of stabilization for the damped wave equation on tori
$$(\partial_t^2 \Delta )u +a(x) \partial _t u =0.$$
When the damping coefficient $a(x)$ is continuous the question is quite well understood and the geometric control condition is necessary and sufficient for uniform (hence exponential) decay to hold. When $a(x)$ is only $L^{\infty}$ there are still gaps in the understanding.
Using second microlocalization we completely solve the question for
Damping coefficients of the form
$$a(x)=\sum_{i=1}^{J} a_j 1_{x\in R_j},$$
Where $R_j$ are cubes.
This is a joint work with P. Gérard
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Data assimilation for high dimensional nonlinear forecasting
David Kelly (New York University, Courant Institute)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The prototypical problem of data assimilation is weather forecasting. The goal is to provide state predictions for nonlinear and extremely high dimensional models, the novelty is to determine how best to guide these predictions using the now vast amounts of observational data that is available to forecasters. Most data assimilation algorithms have been developed within the applied sciences, notably in meteorology. As a consequence these algorithms sit on little to no mathematical foundation, instead they are ad hoc methods whereby wellunderstood algorithms are tweaked to make them more suitable to the given problem. Nevertheless, numerical evidence suggests that these algorithms work  they often possess mathematically well understood properties, such as filter accuracy (tracking the right signal) and stability (ergodicity).
We will discuss two roles that mathematicians can play in this field: 1  to develop understanding of the mathematics behind seemingly ad hoc forecasting algorithms and 2  to use this understanding to propose new algorithms that are strongly backed by theory.
This is based on several joint works with Andy Majda, Andrew Stuart and Xin Tong.
 Supplements



Oct 30, 2015
Friday

09:30 AM  10:30 AM


Some generic features of dynamics in a highdimensional rugged landscape
Jorge Kurchan (École Normale Supérieure)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The organization of high dimensional random `rugged' landscapes
has some universal features, some of them quite antiintuitive to humans used to think in low
dimensions. The dynamics within such landscapes is also quite antiintuitive, for the same reasons.
I will describe how one may learn from analytic solutions of simplified models, and some
conjectures that one would like to prove
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Long wave limit for Schrodinger maps
Pierre Germain (New York University, Courant Institute)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will show how a new class of KdVtype equations can be derived from Schrodinger maps with a constraining potential in the long wave limit. This gives a general framework encompassing long waves limits for GrossPitaevski and LandauLifschitz equations. This is joint work with Frederic Rousset.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Hitting questions and multiple points for stochastic PDE (SPDE) in the critical case
Carl Mueller (University of Rochester)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Hitting questions play a central role in the theory of Markov processes. For example, it is well known that Brownian motion hits points in one dimension, but not in higher dimensions. For a general Markov process, we can determine whether the process hits a given set in terms of potential theory. There has also been a huge amount of work on the related question of when a process has multiple points.
For SPDE, much less is known, but there have been a growing number of papers on the topic in recent years. Potential theory provides an answer in principle. But unfortunately, solutions to SPDE are infinite dimensional processes, and the potential theory is intractible. As usual, the critical case is the most difficult.
We will give a brief survey of known results, followed by a discussion of an ongoing project with R. Dalang, Y. Xiao, and S. Tindel which promises to answer questions about hitting points and the existence of multiple points in the critical case
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


The cubic Dirac equation in $H^\frac12(\R^2)$
Ioan Bejenaru (University of California, San Diego)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Global wellposedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{\frac12}(\R^2)$ is established. The proof is based on a sharp endpoint Strichartz estimate for the KleinGordon equation in dimension $n=2$, which is captured by constructing an adapted systems of coordinate frames. This is joint work with S. Herr.
 Supplements


