



(U Wisconsin, Madison) 
Singular integrals and a problem on mixing flows  We discuss a problem on mixing flows and how it is related to estimates for multilinear singular integral forms of ChristJourn\'e type. This is joint work with Charles Smart and Brian Street.  
(Purdue U) 
Finite time singularity of the nematic liquid crystal flow in 3D  In this talk, I will discuss the initial and boundary value problem for a simplified nematic liquid crystal flow in dimension three, and present two examples of finite time singularity. The first example is constructed within the class of axisymmetric solutions, and the second example is constructed for any generic initial data $(u_0, d_0)$ that has sufficiently small energy, and $d_0$ has a nontrivial topology. This is a joint work with T. Huang, F. H. Lin, and C. Liu.  
(Penn State U) 
Some energy inequalities involving fractional GJMS operators  Fractional GJMS operators are conformally covariant operators with principal symbol a power of the Laplacian which are typically defined on the boundary of an asymptotically hyperbolic manifold via scattering theory. I will describe an equivalent formulation using conformally covariant operators in the interior which yields, for a large class of manifolds, an energy inequality relating the (conformally invariant) energies of the fractional GJMS operators and certain local operators in the interior. As an application, I establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.  
(JHU) 
Selfshrinkers to the mean curvature flow asymptotic to isoparametric cones  In this talk, we show how to construct an end of a selfsimilar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in $\bfR^{n+1}$ an {\em isoparametric cone} if C is the cone over a compact embedded isoparametric hypersurface $\Gamma \subset \bfS^n$. The theory of isoparametic hypersurfaces is extremely rich and there are infinitely many distinct classes of examples, each with infinitely many members. This is joint work with PoYao Chang.  


(Rutgers) 
The Fundamental theorem of Calculus, Generalizations and Applications  We extend the one dimensional fundamental theorem of Calculus to several variables, in particular to Riemannian symmetric spaces and Nilpotent Lie groups. We present applications to NavierStokes in 2D and Magnetism. This is joint work with Polam Yung and Jean Van Schaftingen.  
(U Chicago) 
A Free Boundary Problem for the Parabolic Poisson Kernel  We study a free boundary problem for caloric measure first introduced by Hofmann, Lewis and Nyström. In particular, we show that the oscillation of the Poisson kernel of a domain controls the oscillation of the unit space normal to that domain (this is the parabolic analogue of Kenig and Toro's work on harmonic measure). The key step is classifying "flat blowups." To do so, we adapt the work of Andersson and Weiss on solutions to a parabolic problem arising in combustion.  
(Cambridge U) 
Global nonlinear stability of Minkowski space for the massless EinsteinVlasov system'  Massless collisionless matter is described in general relativity by the massless Einstein–Vlasov system. I will present key steps in a proof that, for smooth asymptotically flat Cauchy data for this system, sufficiently close, in a suitable sense, to the trivial solution, Minkowski space, the resulting maximal development exists globally in time and asymptotically decays appropriately. By appealing to the corresponding result for the vacuum Einstein equations, a monumental result first obtained by Christodoulou–Klainerman in the early ’90s, the proof reduces to a semiglobal problem. A key step is to estimate certain Jacobi fields on the mass shell, a submanifold of the tangent bundle of the spacetime endowed with the Sasaki metric.  
STARTING AT 4:30PM 
(UC Berkeley) 
Dispersive solutions to spherically symmetric EinsteinScalar Field system  The spherically symmetric EinsteinScalar Field system is a simple model for gravitational collapse. A satisfactory qualitative description of the global dynamics is known through a remarkable work by D. Christodoulou: Generically, asymptotically flat data give rise to solutions which are either dispersive (i.e., future causal geodesically complete) or possess a black hole. The aim of this talk is to give a detailed quantitative description of the first generic behavior, namely the dispersive case. I will first present a dichotomy (established in a joint work with J. Luk) which holds for any dispersive solution regardless of the size of the data: Either the fields decay at universal polynomial rates, or a certain local scaleinvariant norm near the axis of symmetry is nonvanishing towards future infinity. Next, I will describe a recent construction (with J. Luk and S. Yang) of solutions with large initial mass and scaleinvariant norms, which fall into the first alternative of the dichotomy. 
(U Kentucky) 
Regularity of $p$ Harmonic Functions  In this talk, given a bounded domain $ \Om \subset \rn{n} = $ Euclidean $ n $ space, we discuss what is known about the interior regularity of a $p$ harmonic function $ u $ in $ \Om, $ when $ 1 < p < \infty, \\ p \not = 2. $ That is, $u: \Om \rar \re $ is a weak solution to the $p$ Laplace equation: \[ \nabla \cdot \left( \nabla u ^{p2} \nabla u \right ) = 0 \mbox{ in $ \Om, $ when } 1 < p < \infty, p \not = 2. \] Included in this discussion will be recent work of Vogel and the author on real analytic solutions to the $p$ Laplace equation.  
(Lehigh) 
Geometry of Ricci Solitons  The concept of Ricci solitons was introduced by R. Hamilton in mid 1980s. Ricci solitons are natural extensions of Einstein manifolds. They are also selfsimilar solutions to the Ricci flow and play an important role as limiting singularity models. In this talk, I shall describe some recent progress on the geometry and classifications of Ricci solitons.  
Extra Seminar (in Krieger 413) 
(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 When the nonlinearity is of the form $F(u)=u^{m1}u$, with $m>1$, global Harnack estimates are the key tool to understand the sharp asymptotic behaviour of the solutions. We finally show that solutions are classical, and even C^{\infty} in space in the interior of the domain, when the operator $\mathcal{L}$ is the (restricted) fractional Laplacian. The above results are contained on a series of recent papers with J. L. Vazquez, and also with A. Figalli, Y. Sire and X. RosOton. 
Extra Seminar 
(U Autonoma De Madrid) 
Archive of Analysis Seminar: