# Seminars & Events for Analysis Seminar

##### Area-Minimizing Surfaces in Asymptotically Flat Three-Manifolds

We will discuss joint work with M. Eichmair in which we show that asymptotically flat three-manifolds with non-negative scalar curvature do not admit unbounded area-minimizing boundaries unless the ambient manifold is flat.

##### Global Existence, Blowup and Scattering for large data Supercritical and other wave equations

**PLEASE NOTE NEW START TIME OF 3:00.** I present a new approach to classify the asymptotic behavior of certain classes of wave equations, supercritical and others, with large initial data. In some cases, as for Nirenberg type equations, a fairly complete classification of the solutions (finite time blowup or global existence and scattering) is proved. New results are obtained for the well known monomials wave equations in the sub, critical and super critical cases. This approach, developed jointly with M. Beceanu, is based on a new decomposition into incoming and outgoing waves for the wave equation, and the positivity of the fundamental solution of the wave equation in three dimensions.

##### Nonuniqueness of weak solutions to the SQG equation

##### Non-linear stability of Kerr-de Sitter black holes

In joint work with András Vasy, we recently proved the stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will explain the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges; I will also address the issue of finding the mass and the angular momentum of the final black hole.

##### Symmetries of linearized Einstein equations on Kerr spacetime

Motivated by the black hole stability problem, we discuss the structure of linear test fields on Kerr spacetime. The dynamics of the linearized gravitational field on a Kerr background is governed by certain curvature components solving the Teukolsky master equations (TME) and Teukolsky-Starobinski identities (TSI). I will show that the TME and TSI operators are self-adjoint in an appropriate sense. Combined with a result due to Wald from 1978, this leads to symmetry operators of the linearized Einstein equations of order four and six, respectively. Interpretations and possible applications will be discussed. The results were derived using spinors and advanced symbolic computer algebra tools for xAct, which I will introduce briefly.

##### Symmetries and Critical Phenomena in Fluids

We describe recent results on studying the dynamics of fluid equations in critical spaces. While it is known that the incompressible Euler equation is ill-posed in the class of Lipschitz velocity fields (even when the data is taken to be smooth away from the origin), we prove well-posedness (global in 2d and local in 3d) for merely Lipschitz data which is smooth away from the origin and satisfies a mild symmetry assumption. To do this requires a deep understanding of the nature of unboundedness of singular integrals on $L^\infty$. After this, we extract a simplified equation which is satisfied by "scale invariant" solutions which lie within the setting of our local well-posedness theory. These scale-invariant solutions, in the 2d Euler setting, can be shown to have very interesting dynamical properties.

##### Global well posedness and scattering for the cubic nonlinear wave equation

In this talk we discuss the defocusing, cubic nonlinear wave equation. We prove scattering for radially symmetric, nearly sharp initial data. We do not assume an a priori bound on the critical norm. We prove this by studying the wave equation in hyperbolic coordinates.

##### Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat data

In this talk, I will present a recent work (joint with Jonathan Luk) on the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it was previously proved (by M. Dafermos and I. Rodnianski) that a certain formulation of the strong cosmic censorship conjecture is *false*, namely, the maximal globally hyperbolic development of a data set in this class is extendible as a Lorentzian manifold with a $C^0$ metric. Our main result is that, nevertheless, a weaker formulation of the conjecture is true for this model, i.e., for a generic data set in this class, the maximal globally hyperbolic development is inextendible as a Lorentzian manifold with a $C^2$ metric.

##### On a problem of Kahane in higher dimensions

**Please note special day, time and room. **We characterise those real analytic mappings from T^k to T^d which map absolutely convergent Fourier series on T^d to uniformly convergent Fourier series via composition.

We do this with respect to rectangular summation on T^k. We also investigate uniform convergence with respect to square sums and highlight the differences which arise.

##### CANCELLED: The Kakeya needle problem for rectifiable sets

* THIS SEMINAR HAS BEEN CANCELLED. *We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is a joint work with Alan Chang.

##### Maximizers for the Stein–Tomas Inequality

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein–Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein–Tomas inequality. Our result is valid in any dimension.

The talk is based on joint work with E. Lieb and J. Sabin.

##### The Kakeya needle problem for rectifiable sets

We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is a joint work with Alan Chang.

##### A singular perturbation problem for the fractional Allen-Cahn equation

I will describe some convergence result for a singular perturbation of the fractional Allen-Chan equation involving powers of the laplacian less than 1/2. In this case, one converges in a suitable sense to stationary minimal nonlocal surfaces that I will describe precisely. The convergence happens to be strong due to a deep result of Geometric Measure Theory due to Marstrand. This is a feature of the non locality of the problem.

##### Channel of energy for outgoing waves and universality of blow up for wave maps

We will introduce a recently found channel of energy inequality for outgoing waves, which has been useful for semi-linear wave equations at energy criticality. Then we will explain an application of this channel of energy argument to the energy critical wave maps into the sphere. The main issue is to eliminate the so-called ``null concentration of energy". We will explain why this is an important issue in the wave maps. Combining the absence of null concentration with suitable coercive property of energy near traveling waves, we show a universality property for the blow up of wave maps with energy that are just above the co-rotational wave maps. Difficulties with extending to arbitrarily large wave maps will also be discussed. This is joint work with Duyckaerts, Kenig and Merle.

##### Stable shock formation for solutions to the multidimensional compressible Euler equations in the presence of non-zero vorticity

It is well-known since the foundational work of Riemann that plane symmetric solutions to the compressible Euler equations may form shocks in finite time. For a class of simple plane symmetric solutions, we prove that the phenomenon of shock-formation is stable under perturbations of the initial data that break the plane symmetry with potentially non-vanishing vorticity. In particular, this is the first constructive shock-formation result for which the vorticity is allowed to be non-vanishing at the shock. We show that the vorticity remains bounded all the way up to the shock, and that the dynamics are well-described by the irrotational compressible Euler equations. This is a joint work with J. Speck (MIT), which is partly an extension of an earlier joint work with J. Speck (MIT), G. Holzegel (Imperial) and W. Wong (Michigan State).

##### Schrodinger equations and irrational tori

We prove long-time Strichartz estimates for solutions to the linear Schrodinger equation on generic irrational tori. This improves the recent work of Bourgain and Demeter. As an application, we also establish

polynomial bounds for Sobolev norm of solutions to the energy critical nonlinear Schrodinger equation in 3D. The first part is joint work with P. Germain and L. Guth.

##### Some free boundary problems for moving drops: Existence and homogenization

We will present two simplified models describing the motion of a small liquid drop on a solid surface: the thin film equation and the quasi-static approximation. Existence and homogenization results for these models will be presented. Of particular interest to us is the description of a drop sliding down an inclined plane.

##### Dispersive estimates for undergraduates

In this extremely elementary seminar, I discuss how to prove some scaling-sharp dispersive estimates for the linear Schrodinger equation using purely physical space methods. The main part of the argument uses only the fundamental theorem of calculus and is accessible to undergraduates, even those not enrolled at Princeton.

##### CANCELLED: Square functions for directional operators on the plane

**PLEASE NOTE THAT THIS SEMINAR HAS BEEN CANCELLED. **

##### Can we compute everything?

It is often desirable to solve mathematical problems as a limit of simpler problems. However, are such techniques always guaranteed to work? For instance, the problem of finding roots of polynomials of degree higher than two was only solved in the 1980s (Newton's method isn't guaranteed to converge)! Doyle and McMullen showed that this is only possible if one allows for multiple independent limits to be taken, not just one. They

called such structures "Towers of Algorithms". In this talk I will apply this idea to other problems (such as computational quantum mechanics, inverse problems, spectral analysis), show that Towers of Algorithms are a