(Imperial) Special Seminar 
Global nonlinear stability of Minkowski space for EinsteinVlasov systems  Global nonlinear stability of Minkowski space for the massive and massless EinsteinVlasov systems. Abstract: The EinsteinVlasov system describes an ensemble of collisionless particles interacting via gravity, as modelled by general relativity. Under the assumption that all particles have equal mass there are two qualitatively different cases according to whether this mass is zero or nonzero. I will present two theorems concerning the global dispersive properties of small data solutions in both cases. The massive case is joint work with Hans Lindblad.  
(UC Santa Barbara) 
Minmax theory for constant mean curvature (CMC) hypersurfaces  In this talk, I will present constructions of closed CMC hypersurfaces using minmax method. In particular, given any closed Riemannian manifold, I will show the existence of closed CMC hypersurfaces of any prescribed mean curvature. This is a joint work with Jonathan Zhu.  
(Yale U) 
A variational approach to axially symmetric Maxwell perturbations of Kerrde Sitter black holes  The Kerrde Sitter black hole family is a solution of Einstein's equations of general relativity with a positive cosmological constant. After reviewing some background on these spacetimes, we shall discuss the proof that there exits a phase space of canonical variables for the (unconstrained) axially symmetric Maxwell's equations propagating on Kerrde Sitter, such that their motion is restricted to the level sets of a positivedefinite Hamiltonian, despite the ergoregion. If time permits, we shall discuss the equivalent results for the corresponding fully coupled (and constrained) EinsteinMaxwell initial value problem.  
(UC Santa Cruz) 
Hypersurfaces in hyperbolic space  In this talk I will report our recent works on convex hypersurfaces in hyperbolic space. I will talk on nonnegatively curved immersed as well as Ricci nonnegative embedded hypersurfaces in hypersurfaces in hyperbolic space.  
(UC Berkeley) 
Nonlinear stability of Kerrde Sitter black holes  I will explain some ideas behind the proof of the global stability of the Kerrde Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations when the cosmological constant is positive. 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 gauge, within a carefully chosen finitedimensional family of gauges, in which we can find the global solution. I will also address the issue of finding the mass and the angular momentum of the final black hole. This talk is based on joint work with András Vasy.  
(JHU) 
Hausdorff stability of round spheres under smallentropy perturbation  ColdingMinicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. BernsteinWang proved that round spheres $\mathbb S^n$ minimize entropy among all closed hypersurfaces for $n\leq6$, and the result is generalized to all dimensions by Zhu. BernsteinWang later also proved that the round 2sphere is actually Hausdorff stable under smallentropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyperspheres in all dimensions.  
(Ryukoku U) 
An exterior nonlinear elliptic problem with a dynamical boundary condition  We consider the nonnegative solution of a semilinear elliptic equation with a dynamical boundary condition. Several results on existence, nonexistence and largetime behavior of small solutions were obtained before for the halfspace case. In this talk we study the effects of the change of the domain from the halfspace to the exterior of the unit ball. We obtain the critical exponent with respect to the existence of solutions and the decay rate of small globalintime solutions. Furthermore, we show that local solvability is equivalent to global solvability. This talk is based on the joint work with M. Fila (Comenius Univ.) and K. Ishige (Tohoku Univ.).  
(Minnesota) 
On Regularity and Longtime Behavior of 1D Models  In the 1980s, P. Constantin, P. Lax, and A. Majda introduced a 1d equation which models some of the features of the 3d incompressible Euler equation, showed that the model equation is explicitly solvable, and has solutions which can develop singularities from smooth data. Introducing a natural transport term into the original model, as proposed by De Gregorio in the 1990s, seems to have a regularizing effect, quite visible in numerical simulations. The definite answer to the regularity problem for the De Gregorio model is unknown at this time. I will discuss some recent partial results concerning these topics. The De Gregorio equation has some interesting quasilinear dispersive features.  
(Northwestern) 
On minimizers and critical points for anisotropic isoperimetric problems  Anisotropic surface energies are a natural generalization of the perimeter that arise in models for equilibrium shapes of crystals. We discuss some recent results for anisotropic isoperimetric problems concerning the strong quantitative stability of minimizers, bubbling phenomena for critical points, and a weak Alexandrov theorem for nonsmooth anisotropies. Part of this talk is based on joint work with Delgadino, Maggi, and Mihaila.  
(U Chicago) 
An infinite sequence of conserved quantities for the cubic GrossPitaevskii hierarchy on R  We consider the (de)focusing cubic GrossPitaevskii (GP) hierarchy on R, which is an infinite hierarchy of coupled linear nonhomogeneous PDE which appears in the derivation of the cubic nonlinear Schrodinger (NLS) equation from quantum manyparticle systems. Motivated by the fact that the cubic NLS on R is an integrable equation which admits infinitely many conserved quantities, we exhibit an infinite sequence of operators which generate analogous conserved quantities for the GP hierarchy. This is joint work with Andrea Nahmod, Natasa Pavlovic, and Gigliola Staffilani.  
(U Missouri) 
Rectifiability, harmonic measure, and boundary behavior of harmonic functions  We discuss recent progress in an ongoing program to understand the relationship between rectifiability of the boundary of a domain, and the boundary behavior of harmonic functions.  
(U Autonoma de Madrid) 
Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains  We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L} F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the three most common versions of the fractional Laplacian $(\Delta)^s$, $0 First we present some result about sharp boundary behaviour and regularity for the associated stationary elliptic problem (semilinear). Next, we will shortly present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions. We will devote special attention to the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and (sharp) asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our techniques cover also the local case s = 1 and provide new results even in this setting. A surprising instance of this problem is the possible presence of nonmatching powers for the boundary behavior: for instance, when $\mathcal{L}=(\Delta)^s$ is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that, whenever $2s > 1  1/m$, solutions behave as $dist^{1/m}$ near the boundary; on the other hand, when $2s \le 1  1/m$, different solutions may exhibit different boundary behaviors even for large times. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic case, for which we will shortly present the most recent results. The above results are contained on a series of recent papers in collaboration with A. Figalli, Y. Sire, X. RosOton and J. L. Vazquez. 

(Imperial) 
The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge  The Schwarzschild solution, discovered in 1916, describes a black hole spacetime in Einstein’s theory of relativity. Its stability as a solution to the Einstein vacuum equations is fundamental to the black hole notion. In a recent work [2016] Dafermos, Holzegel and Rodnianski proved the linear stability of the Schwarzschild solution via an analysis of the nulldecomposed Bianchi equations coupled to the null structure equations. In this talk we shall prove that the Schwarzschild solution is in fact linearly stable as a solution to the vacuum Einstein equations when expressed in a generalised wave gauge. The two approaches should be compared with the monumental proof of the nonlinear stability of the Minkowski space by Christodoulou and Klainerman and the later complementary proof by Lindblad and Rodnianski which employed wave coordinates. The proof proceeds by exploiting a gauge freedom present in the linearised theory which culminates in the observation that gaugeinvariant quantities satisfy the decoupled scalar wave equations described by the Regge—Wheeler and Zerilli equations. The analysis of these equations builds on earlier fundamental work by Dafermos and Rodnianski for the scalar wave equation on the Schwarzschild spacetime which shall be reviewed. 
Archive of Analysis Seminar:
Spring 2017 Fall 2016 Spring 2016 Fall 2015 Spring 2015 Fall 2014 Spring 2014 Fall 2013