Date	Speaker	Title	Abstract
1/25	NO SEMINAR
2/1	NO SEMINAR
2/8	Andreas Seeger (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 Christ-Journ\'e type. This is joint work with Charles Smart and Brian Street.
2/15	Changyou Wang (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.
2/22	Jefferey Case (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.
2/29	Joel Spruck (JHU)	Self-shrinkers to the mean curvature flow asymptotic to isoparametric cones	In this talk, we show how to construct an end of a self-similar 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 Po-Yao Chang.
3/7	No Seminar
3/21	Sagun Chanillo (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 Navier-Stokes in 2D and Magnetism. This is joint work with Po-lam Yung and Jean Van Schaftingen.
3/28	Max Engelstein (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.
4/4	Martin Taylor (Cambridge U)	Global nonlinear stability of Minkowski space for the massless Einstein--Vlasov 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 semi-global 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.
4/11 STARTING AT 4:30PM	Sung-Jin Oh (UC Berkeley)	Dispersive solutions to spherically symmetric Einstein-Scalar Field system	The spherically symmetric Einstein-Scalar 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 scale-invariant norm near the axis of symmetry is non-vanishing towards future infinity. Next, I will describe a recent construction (with J. Luk and S. Yang) of solutions with large initial mass and scale-invariant norms, which fall into the first alternative of the dichotomy.
4/18	John Lewis (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: \[ \nabla \cdot \left( |\nabla u |^{p-2} \nabla u \right ) = 0 \mbox{ in $ \Om, $ when } 1 < p < \infty, p \not = 2. \] Included in this discussion will be recent work of Vogel and the author on real analytic solutions to the $p$ Laplace equation.
4/25	Huai-Dong Cao (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 self-similar 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.
4/26 (2-3pm) Extra Seminar (in Krieger 413)	Matteo Bonforte (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 $(-\Delta)^s$, $01$\,. We will 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 then show sharper lower and upper estimates up to the boundary, which fairly combine into various forms of Harnack type inequalities. The standard Laplacian case $s=1$ is included and the linear case $m=1$ can be recovered in the limit in most of the results. When the nonlinearity is of the form $F(u)=|u|^{m-1}u$, 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. Ros-Oton.
5/4 Extra Seminar	Juan-Luis Vazquez (U Autonoma De Madrid)

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