Abstract: We consider the T^4 cubic NLS which is energy-critical. We study the unconditional uniqueness of solution to the NLS via the cubic Gross-Pitaevskii hierarchy, an uncommon method, and does not require the existence of solutions in Strichartz type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates, which fail on T^4. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel-Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the H^1 unconditional uniqueness for the NLS under the infinite hierarchy framework. This work establishes a unified scheme to prove H^1 uniqueness for the R^3/R^4/T^3/T^4 energy-critical Gross-Pitaevskii hierarchies and thus the corresponding NLS.
Abstract: The tools from machine learning (ML) offer exciting new prospects in scientific computing in general, and the numerical solution of PDEs in particular. In this talk, I will discuss some recent results in this context, by focusing on problems with a variational formulation, and discussing three main ingredients of their solution by ML: (i) approximation quality, i.e. how accurate the representation of the solution by a neural network can in principle be; (ii) optimization, i.e. how effective are the methods to train the parameter of the network; and (iii) generalization error, i.e. how much data is necessary to obtain a solution accurate also outside this data set. In particular I will show that this third aspect typically requires using importance sampling methods for data acquisition. These results will be illustrated on eigenvalue problems in high dimension. This is joint work with Grant Rotskoff.
The seminar will be available on zoom at the link https://umd.zoom.us/j/98038768667?pwd=L3lrQ0lZVmQyM1ZQLzA0dkp3OUJXUT09
Abstract: In this talk, we are interested in studying linear stability properties of perturbations around the Couette flow with constant density for an isentropic compressible fluid.ÃÂ In the inviscid case, we show a generic Lyapunov type instability for the density and the irrotational part of the velocity, namely their L^2 norm grows asÃÂ t^(1/2). Instead, the solenoidal component of the velocity experience inviscid damping, meaning that it decays to zero even in the absence of viscosity.ÃÂ For a viscous compressible fluid, we show that the perturbations decay exponentially fast on a faster time-scale with respect to the standard diffusive one, a phenomenon also called enhanced dissipation. This is a joint work with P. Antonelli and P. Marcati.ÃÂ ÃÂ
Abstract: In the recent years, several major results have been obtained in the problem of finding a constant Q-curvature metric in a given conformal class in dimensions bigger than 5. This talk will cover new results concerning existence and multiplicity of such metrics. I will first present a rather general geometric approach to prove existence and multiplicity of regular metrics, giving several explicit examples. Then I will move to the case of singular metrics, i.e. complete metrics with constant Q-curvature outside of a closed set. This requires to develop several tools to handle 4th order equations (but applicable actually to higher order ones). I will also provide some explicit examples of such metrics and investigate their multiplicity. I will state open problems as well.
Abstract: In this talk I will discuss a recent work which proves stability of Prandtl's boundary layer in the vanishing viscosity limit. The result is an asymptotic stability result of the background profile in two senses: asymptotic as the viscosity tends to zero and asymptotic as x (which acts a time variable) goes to infinity. In particular, this confirms the lack of the "boundary layer separation" in certain regimes which have been predicted to be stable. This is joint work w. Nader Masmoudi (Courant Institute, NYU).
Abstract: A point charge is a particularly basic and important equilibrium of the Vlasov-Poisson equations, and the study of its stability has inspired several major contributions. In this talk we present some recent work, which brings a fresh perspective on this problem. Our new approach combines a Lagrangian analysis of the linearized problem with an Eulerian PDE framework in the nonlinear analysis, all the while respecting the symplectic structure. As a result, for the case of radial initial data, we see that solutions are global and in fact disperse to infinity via a modified scattering along trajectories of the linearized flow. This is joint work with Benoit Pausader (Brown University).
Abstract: The 1d cubic nonlinear SchrÃÂ¶dinger equation (NLS) and the modified Korteweg-de Vries equation (mKdV) are two of the most intensively studied nonlinear dispersive equations. Not only are they important physical models, arising, for example, from the study of fluid dynamics and nonlinear optics, but they also have a rich mathematical structure: they are both members of the ZS-AKNS hierarchy of integrable equations. In this talk, we discuss an optimal well-posedness result for the cubic NLS and mKdV on the line. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, which in turn rests on the discovery of a one-parameter family of microscopic conservation laws. This is joint work with Rowan Killip and Monica ViÃÂan.
Abstract: In this talk, I will give an overview of some of what is known about solutions to the thin obstacle problem, and then move on to a discussion of a higher regularity result on the singular part of the free boundary. This is joint work with Xavier Fernandez-Real.
Abstract: Anisotropic energies are functionals defined by integrating over a generalized surface (such as a current or a varifold) an integrand depending on the tangent plane to the surface. In the case of a constant positive integrand, one obtains the area functional, and hence one can see anisotropic energies as a generalization of it. A long standing question in geometric measure theory is to establish regularity properties of critical points to such functionals. In this talk, I will discuss some recent developments on this theory, addressing in particular the question of rectifiability of stationary points and regularity of stationary Lipschitz graphs. The talk is based on joint work with Antonio De Rosa.
Abstract: Constant Mean Curvature (CMC) surfaces constitute a classical subject in Differential Geometry and are mathematical models in many disciplines of science. In this talk, I will present a recent work on the existence of CMC 2-spheres in an arbitrary Riemannian 3-sphere. This is a joint work with Da Rong Cheng.
Abstract: The self-dual U(1)-Yang-Mills-Higgs functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly in the 2-dimensional and Kaehler settings) are objects of long-standing interest in low-dimensional gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds. We will also discuss recent work with Davide Parise and Alessandro Pigati developing the associated Gamma-convergence machinery, and describe some geometric applications.
Abstract: The N-membrane problem is the study of shapes of elastic membranes being pushed against each other. The main questions are the regularity of the functions modeling the membranes, and the regularity of the contact regions between consecutive membranes. These are classical questions in free boundary problems. However, very little is known when N is larger than 2. In this case, there are multiple free boundaries that cross each other, and most known techniques fail to apply. In this talk, we discuss, for general N, the optimal regularity of the solutions in arbitrary dimensions, and a classification of blow-up solutions in 2D. Then we focus on the regularity of the free boundaries when N=3. This talk is based on two recent joint works with Ovidiu Savin (Columbia University).
Abstract: We discuss recent work on some quasilinear Hamiltonian dispersive equations of KdV and NLS-type that act as toy models for the phenomenon of degenerate dispersion, where the dispersion relation may degenerate at a point in space. We will cover some special stationary and traveling solutions, as well as new results on the functional framework for time evolution for these equations. This connects nicely to results on quasilinear equations that I have worked on in a series of works with Jason Metcalfe and Daniel Tataru, as well as to recent interesting work of Sung-Jin Oh and In-Jee Jeong on the magnetohydrodynamic equations. This is joint work with Pierre Germain and Ben Harrop-Griffiths.
Abstract: A celebrated theorem of Jorgens-Calabi-Pogorelov says that global convex solutions to the Monge-Ampere equation det(D^2u) = 1 are quadratic polynomials. On the other hand, an example of Pogorelov shows that local solutions can have line singularities. It is natural to ask what kinds of singular structures can appear in functions that solve the Monge-Ampere equation outside of a small set. We will discuss examples of functions that solve the equation away from finitely many points but exhibit polyhedral and Y-shaped singularities. Along the way we will discuss geometric and applied motivations for constructing such examples, as well as their connection to a certain obstacle problem for the Monge-Ampere equation.
Abstract: In this talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. In particular, an example of such a system of bosons leads to a quintic nonlinear Schrodinger equation, which we rigorously derived in a joint work with Thomas Chen. An example of a system of classical particles that allows instantaneous ternary interactions leads to a new kinetic equation that can be understood as a step towards modeling a dense gas in non-equilibrium. We call this equation a ternary Boltzmann equation and we rigorously derived it with Ioakeim Ampatzoglou. Time permitting, we will also discuss the recent work with Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this work introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time.
Abstract: Let M be a compact 3-manifold with scalar curvature at least 1. We show that there exists a Morse function f on M, such that every connected component of every fiber of f has genus, area and diameter bounded by a universal constant. The proof uses Min-Max theory and Mean Curvature Flow. This is a joint work with Davi Maximo. Time permitting, I will discuss a related problem for macroscopic scalar curvature in metric spaces (joint with Boris Lishak, Alexander Nabutovsky and Regina Rotman).
Abstract: Topological insulators are phases of matter that act like extraordinarily stable waveguides along their boundary. They have a rich mathematical structure that involves PDEs, spectral theory, and topology. I will survey some results and discuss some (semi-classical) directions of research.
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