(Georgia Tech) 
Total curvature and the isoperimetric inequality: Proof of the CartanHadamard conjecture  The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we show that this inequality also holds in simply connected spaces of nonpositive curvature, known as CartanHadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller. The proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for the smooth approximation of the signed distance function, via infconvolution and Reilly type formulas among other techniques. Immediate applications include sharp extensions of Sobolev and FaberKrahn inequalities to spaces of nonpositive curvature. This is joint work with Joel Spruck. ssss  
(U. Penn) 
A trajectory map for the pressureless Euler equations  We consider the dynamics of a collection of particles that interact pairwise and are restricted to move along the real line. Moreover, we focus on the situation in which particles undergo perfectly inelastic collisions when they collide. The equations of motion are a pair of partial differential equations for the particles’ mass distribution and local velocity. We show that solutions of this system exist for given initial conditions by rephrasing these equations in Lagrangian coordinates and then by solving for the associated trajectory map.  
(JHU) 
Self shrinkers and low entropy mean curvature flow with surgery  In this talk I’ll explain the mean curvature flow with surgery for mean convex hypersurfaces of R^n under a low entropy assumption, and how to adapt that to study self shrinkers of low entropy. This is a joint work with Shengwen Wang.  
(IAS) 
Square function estimate for the cone in R^3.  We prove a sharp square function estimate for the cone in R^3 and consequently the local smoothing conjecture for the wave equation in 2+1 dimensions. The proof uses an incidence estimate for points and tubes and induction on scales. This is joint work with Larry Guth and Ruixiang Zhang.  


(MIT) 
Eigenfunctions on selfsimilar solutions to geometric flows  Eigenfunctions of the linearized operator of geometric flow on selfsimilar solutions determine the asymptotic behavior. We will consider the eigenfunctions of the Jacobi operator to the mean curvature flow over the round sphere and cylinder. And will see how they determine the asymptotic shape of asymptotically cylindrical ancient flow, marriage ring singularity as well as the regularity of the arrival time. Also, we would discuss about further application to the nonlinear flows including the Gauss curvature flow.  
(U. Maryland) 
Schrodinger solutions on a fractal set  If a function has Fourier support on the truncated paraboloid, then what can we tell about its behavior on a set of a lower dimension in physical space? This question relates to several problems in PDEs and geometric measure theory. We'll see some special Schrodinger solutions, and talk about various results derived from decoupling and induction on scales, which allow us to control Schrodinger solutions on a sparse and spreadout set. Part of this talk is based on joint work with Larry Guth, Xiaochun Li, and Ruixiang Zhang.  
(Princeton) 
Positive scalar curvature and the dihedral rigidity conjecture  In 2013, Gromov proposed a dihedral rigidity conjecture, aiming at establishing a geometric comparison theory for metrics with positive scalar curvature. The conjecture states that if a Riemannian polyhedron has nonnegative scalar curvature in the interior, and weakly mean convex faces, then the dihedral angle between adjacent faces cannot be everywhere less than the corresponding Euclidean model. I will prove this conjecture for a large collection of polytopes. The strategy is to relate this conjecture with a geometric variational problem of capillary type, and apply the SchoenYau minimal slicing technique for manifolds with boundary.  
(Beijing Normal U) 
Optimal boundary regularity for fast diffusion equations in bounded domains  We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvaturelike quantity.  
(Bologna) 
SobolevPoincaré inequality for differential forms in Heisenberg groups  In this talk we prove contact Poincar\'e and Sobolev inequalities in Heisenberg groups $\mathbb H^n$, where the word ``contact'' is meant to stress that de Rham's exterior differential is replaced by the ``exterior differential’’ $d_c$ of the socalled Rumin's complex $(E_0^\bullet,d_c)$. A crucial feature of Rumin’s construction is that $d_c$ recovers the scale invariance of the ``exterior differential’’ $d_c$ under the group dilations associated with the stratification of the Lie algebra of $\mathbb H^n$. These inequalities provide a natural extension of the corresponding usual inequalities for functions in $\mathbb H^n$ and are a quantitative formulation of the fact that $d_c$closed forms are locally $d_c$exact.  
(TennesseeKnoxville) 
Concavity of the arrival time and applications to curve shortening flow  In a breakthrough paper, X.J. Wang in 2011, proved that any compact, convex, ancient solution to mean curvature flow either sweeps out the whole space or lies in a slab region. His result is based on showing that the timeofarrival function for such a solution is concave. In this talk we will show that this concavity property holds for a large class of flows and we will present a connection between this property and Harnack inequalities for hypersurface flows. Finally, we will use Wang's dichotomy theorem to provide a new geometric proof of the classification of convex ancient solutions to curve shortening flow, shown by Daskalopoulos, Hamilton and Sesum, and extend this classification to the noncompact case. This work is partly joint with Langford and partly with Langford and Tinaglia.  
(NYU) 
Quadratic NLS with potential  In this talk some recent results on the longtime behavior of dispersive equa tions with potentials will be presented. We will focus on the NLS equation in 3D with quadratic nonlinearity and time dependent potential. This equation is of interest since it constitutes a good model for the linearization of dispersive equations around special solutions. The method described consists in bringing together the spacetime res onance theory of Germain, Masmoudi and Shatah with tools used in the study of the linear Schrodinger equation.  
(Bologna) 
Schauder estimates at the boundary in Carnot groups  We present a joint work with A. Baldi and G.Cupini on Schauder estimates at the boundary for subLaplacian type operators in Carnot groups. Up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison , is based on the Fourier transform and cannot be repeated in general Lie groups. In this paper we introduce a new method, which allows to build a Poisson kernel starting from the fundamental solution, and to deduce the estimates. 
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