(Huazhong U) 
L^{p} resolvent estimates for simply connected manifold of constant curvature.  Abstract  
(JHU) 
Global well  posedness and scattering for the focusing, energy  critical NLS problem in dimension four.  In this talk we will discuss global well  posedness and scattering for the focusing energy  critical NLS problem in four dimensions. We prove scattering for data below the energy threshold.  
(JHU) 
A sharp lower bound on the entropy of closed hypersurfaces up to dimension six  The entropy of a hypersurface is the supremum of the Gaussian surface area of all translates and scalings of the hypersurface and is a measure of its complexity. It is always greater than one, with equality only for hyperplanes. I will discuss how to use weak mean curvature flows to show that, in low dimensions, the entropy of closed — i.e., compact and without boundary — hypersurfaces is bounded below by the entropy of the round sphere with equality only for the round spheres; proving a conjecture of ColdingIlmanenMinicozziWhite. This is joint work with Lu Wang.  
(JHU) 
Energy Critical EinsteinWave Map System  Wave maps are maps from a Lorentzian manifold to a Riemannian manifold which are critical points of a Lagrangian which is a natural geometrical generalization of the linear wave Lagrangian and the Dirichlet energy of harmonic maps. Selfgravitating wave maps are those from an asymptotically flat Lorentzian manifold which evolves according to Einstein’s equations of general relativity with the wave map itself as the source. The $\text{H}^1$ energy of wave maps is scale invariant if the domain manifold is 2+1 dimensional, hence it is referred to as the critical dimension. Apart from a purely mathematical interest in a natural geometric evolution equation, the motivation to study critical selfgravitating wave maps is that they are naturally connected to the black hole stability conjecture. In this talk, after a brief discussion on the background and formulation of the Cauchy problem of critical selfgravitating wave maps, we shall present a recent proof of the global existence of critical equivariant selfgravitating wave maps before pointing out possible future directions. 

(Princeton U) 
Genus bounds for minmax limits  The minmax process of AlmgrenPitts (and refined by SmithSimon) allows one to do Morse theory on the space of surfaces in a 3manifold in order to construct embedded minimal surfaces. In the 80s Pitts and Rubinstein made a number of conjectures about the geometry and topology of minimal surfaces arising in this fashion. We will give a proof of one of these claims regarding how minimaxing sequences can degenerate. As a corollary we obtain new genus bounds for these surfaces. We will also explain some of the potential applications of minmax minimal surfaces to 3manifold topology.  
(Florida Atlantic U) 
Growth of solutions to the minimal surface equation over unbounded domains  Consider a minimal surface which is the graph of a function over an unbounded domain. If the function is positive and vanishes on the boundary of the domain, what can be said about its growth? I will discuss examples and estimates for growth in terms of the geometry of the domain. We will see an interesting phase transition in the growth constraints for minimal graphs over domains (a) contained in a halfplane as compared to (b) containing a halfplane. This is joint work with Allen Weitsman.  
(Lehigh U) 
Critical point equation problem on 4dimensional compact manifolds.  In the middle of 1980s it has been conjectured that the critical points of the total scalar curvature functional restricted to the set of smooth Riemannian structures of volume 1 and constant scalar curvature are Einstein. This problem is known as CPE Conjecture. In the last decades many authors have been tried to settle up this conjecture, but only partial results were achieved. In this talk, we focus our attention for 4dimensional manifolds. In fact, we shall show that the CPE conjecture is true for 4dimensional half conformally flat manifolds. Moreover, we comment further partial answers to this problem.  
(MIT) CANCELED 
SmallData Shock Formation in Solutions to $3D$ Quasilinear Wave Equations 
I will describe my recent monograph on the formation of shocks, starting from small smooth initial conditions, in solutions to two classes of quasilinear wave equations in $3$ spatial dimensions. The main result states that within the two classes, a \emph{necessary and sufficient} condition for smalldata shockformation is the failure of S. Klainerman's classic null condition. The monograph provides a detailed geometric description of the dynamics from $t=0$ until the first shock. I will highlight some of the main ideas behind the proof including the critical role played by geometric decompositions based on true characteristic hypersurfaces. Some aspects of this work are joint with G. Holzegel, S. Klainerman, and W. Wong. An important precursor is the remarkable result of D. Christodoulou, who in 2007 proved an analogous shockformation result for irrotational regions of smalldata solutions to the relativistic Euler equations. The result was later extended by ChristodoulouMiao to apply to the nonrelativistic Euler equations. Our work recovers both of these results as special cases and generalizes previous smalldata singularityformation results of F. John, S. Alinhac, and many others. 

(U Marseille) 
On a fractional version of a conjecture by De Giorgi and nonlocal minimal surfaces  I will describe several results on symmetry of solutions for semilinear elliptic equations in a connection with a conjecture by De Giorgi. The approach is based on the study of the geometry of the level sets of the solutions and connections are made with nonlocal minimal surfaces, a new type of minimal surface recently introduced. There are many open problems that I will describe.  
(U New Mexico) 
Strichartz and refined local smoothing estimates for wave equations in strictly concave domains  In this talk, we will introduce a family of energy estimates for solutions to the wave and Schrodinger equations in domains with a strictly concave boundary satisfying homogeneous boundary conditions. They can be thought of as refinements of the classical local energy and local smoothing estimates. The estimates show that for frequency localized solutions, taking the square integral of the solution over small frequencydependent collars of the boundary results in a stronger gain in regularity than would be expected for collars of a uniform size. We will also discuss the consequences for the development of Strichartz estimates with subcritical exponents in such domains, in particular providing an avenue for treating Neumann conditions.  
(Stanford U) 
Slowly converging Yamabe flows  We will discuss examples of Yamabe flows which converge polynomially (rather than exponentially) to a metric of constant scalar curvature. We will also explain how the Lojasiewicz–Simon inequality implies that this is the worst that can happen, and explain why in most cases the convergence will actually be exponential. This is joint work with Alessandro Carlotto and Yanir Rubinstein.  
(St. Mary's) 
Pseudospherical Surfaces of Low Differentiability  We will bring together several new ideas in order to classify some $C^1$
surfaces $f$ in $\mathbb{R}^3$ with $K=1$. We consider various special
circumstances including:

Archive of Analysis Seminar: