# Seminars & Events for Analysis Seminar

##### Hidden symmetries and decay for fields outside a Kerr black hole

The Kerr solutions to Einstein's equations describe rotating black holes. For the wave equation in flat-space and outside the non-rotating, Schwarzschild black holes, one method for proving decay is the vector-field method, which uses the energy-momentum tensor and vector-fields. Outside the Schwarzschild black hole, a key intermediate step in proving decay involved proving a Morawetz estimate using a vector-field which pointed away from the photon sphere, where null geodesics orbit the black hole. Outside the Kerr black hole, the photon orbits have a more complicated structure. By using the hidden symmetry of Kerr, we can replace the Morawetz vector-field by a fifth-order operator which, in an appropriate sense, points away from the photon orbits.

##### Narcissistic Skyrmions

The Skyrme model is a nonlinear sigma-model whose topologically non-trivial target space allows the existence of so-called topological solitons. Such solitons were proposed by Skyrme to model nuclear matter. In this talk I will review the history of the model and present recent work with Gary Gibbons and Willie Wong ruling out the existence of solitons with a planar reflection symmetry. In particular, I will show that a Skyrme soliton is attracted to its mirror image, preventing a static equilibrium.

##### Optimal Error Estimates in Stochastic Homogenization

Joint Analysis/PACM Colloquium

##### A limiting interaction energy for Ginzburg-Landau vortices

This is a joint work with Etienne Sandier where we study minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields $H_{c1}$ and $H_{c2}$. In that regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. Among lattice configurations the hexagonal lattice is the unique minimizer of $W$, thus providing a first rigorous hint at the Abrikosov lattice. I will describe briefly how $W$ also appears in the study of the statistical mechanics of Coulomb gases/random matrices.

##### Charge screening in quantum crystals

Density Functional Theory (DFT) has become a major tool in chemistry, materials science, molecular biology and nanotechnology. It is also an inexhaustible source of exciting mathematical and numerical issues. In this talk, I will present some of the variational models derived from DFT, and discuss their mathematical properties. I will focus in particular on the modeling of crystals with local defects. I will show that local defects are always neutral in the Thomas-Fermi-von Weizsäcker (TFW) theory of crystals. In this respect, all crystals therefore behave as metals. The situation is more complex for the Hartree model, which is able to describe not only metals, but also insulators and semi-conductors. In the latter cases, two different values for the charge of a local defect are obtained, depending on the viewpoint adopted.

##### Asymptotic Behavior of Spacetimes Approaching a Schwarzschild solution

Consider a spacetime which approaches a Schwarzschild solution. We will discuss the following problem: Assuming decay of appropriate norms of the Ricci rotation coefficients and their derivatives, can one prove boundedness/ decay for the curvature components and their derivatives? The talk will give a positive answer to this question and explain some of the difficulties arising from the fact that not all curvature components decay. As an important ingredient, we generalize recent work of Dafermos and Rodnianski regarding decay for the wave equation to the setting of the Bianchi equations.

##### Global wellposedness and scattering for the inhomogeneous fourth-order Schrodinger equation

Fourth-order Schrödinger equations were proposed as a correction to the standard model for propagation of laser in nonlinear media and have since appeared in different contexts. In this talk, I will consider the inhomogeneous mass-critical fourth-order Schrödinger equation $iu_t+D^2u-Du+|u|^{8/n}u=0$ and prove global existence and scattering in $L^2$ in high dimensions. The main analysis is reduced to a good understanding of the scaling limit problems which are scale invariant. This is a joint work with Shuanglin Shao.

##### Well-posedness theory for compressible Euler equations in a physical vacuum

An interesting problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. A particular interest is so called physical vacuum which naturally arises in physical problems. The main difficulty lies in the fact that the physical systems become degenerate along the boundary. I'll present the well-posedness result of 3D compressible Euler equations for polytropic gases. This is a joint work with Nader Masmoudi.

##### Normal form-type arguments in the study of dispersive PDEs

Bourgain used normal form reduction and the I-method to prove global well-posedness of one-dimensional periodic quintic NLS in low regularity. In this talk, we discuss the basic notion of normal form reduction for Hamiltonian PDEs and apply it to one-dimensional periodic NLS with general power nonlinearity. Then, we combine it with the "upside-down" I-method to obtain upperbounds on growth of higher Sobolev norms of solutions. In the case of cubic NLS, we explicitly compute the terms arising in the first few iterations of normal form reduction to improve the result. If time permits, we also discuss how one can use a normal form-type argument to prove unconidtional uniqueness of the periodic mKdV in $H^{1/2}$.

##### Linear PDEs in critical regularity spaces: Hierarchical construction of their nonlinear solutions

We construct uniformly bounded solutions of the equations $div(U)=f$ and $curl(U)=f$, for general $f$ in the critical regularity spaces $L^d(R^d)$ and, respectively, $L3(R3)$. The study of these equations was motivated by recent results of Bourgain & Brezis. The equations are linear but construction of their solutions is not. Our constructions are, in fact, special cases of a rather general framework for solving linear equations, $L(U)=f$, covered by the closed range theorem. The solutions are realized in terms of nonlinear hierarchical representations, $U=sum(u_j)$, which we introduced earlier in the context of image processing. The $u_j$'s are constructed recursively as proper minimizers, yielding a multi-scale decomposition of the solutions $U$.

##### Some Inverse problems on Riemann surfaces

We show how to identify a potential $V$ or a connection $\nabla^X=d+iX$ up to gauge on a complex vector bundle from boundary measurements (Cauchy data on the boundary) on a fixed Riemann surface with boundary. This problem consists in showing the injectivity of the nonlinear map $V\to N_V$ (or $X\to N_X$) where $N_V$ and $N_X$ are the Dirichlet-to-Neumann operators associated to the elliptic operator $P=\Delta+V$ or $P=(\nabla^X)^*\nabla^X$. The proof, following ideas of Bukhgeim, is based on the construction of particular complex geometric optics solutions $u=e^{\Phi(z)/h}(1+remainder)$ of $Pu=0$ with holomorphic phases $\Phi$ having isolated critical points.

This is joint work with L.Tzou (Helsinki & MSRI).

##### Periodic DNLS: weighted Wiener measures, gauge transformation and almost global well-posedness

##### Explicit formula for the solution of the cubic Szego equation on the real line and its applications

In this talk we consider the cubic Szego equation: $i u_t = Pi (|u|^2u)$ on the real line, where $Pi$ is the Szeg? projector on non-negative frequencies. This equation was introduced as a model of a non-dispersive Hamiltonian equation. Like 1-d cubic NLS and KdV, it is known to be completely integrable in the sense that it possesses a Lax pair structure. As a consequence, it turns out that a whole class of finite dimensional manifolds, consisting of rational functions, is invariant under the flow of the Szeg? equation.

##### Invertibility of random matrices and applications

Joint Analysis Seminar and PACM Colloquium

##### Product Formulas for Measures and Applications to Analysis

Joint Analysis Seminar and PACM Colloquium

We will discuss elementary product formalisms for positive measures. These appeared in analysis for purposes of examining "harmonic measures" related to elliptic equations (work of R. Fefferman, J. Pipher, C. Kenig).

##### Global dynamics for the ion equation in the Euler-Poisson system

We prove that small perturbations of a constant background in the Euler-Poisson equation for the ions lead to global smooth solutions. This is a manifestation of the stabilization effect of the electric field as the corresponding result does not hold for the closely related compressible Euler equation.

##### Resonances and null structures in weakly dispersive equations

We show a result about global existence of small solutions for a class of scattering critical nonlinear dispersive equations. The problem we consider originates from a classical result of S. Klainerman on the wave equation. Our proof is based on the method of space-time resonances, recently introduced by Germain, Masmoudi and Shatah. In particular, we focus on the relation between resonances and null structures. We will also discuss some related application to scattering critical equations of Schrodinger type.