



(UMD) 
Computing optimal transport maps via optimization 
In joint work with M. Lindsey we rephrase the optimal transportation problem with quadratic costvia a MongeAmpere equationas an infinitedimensional optimization problem, which is often a convex problem. This leads us to define a natural finitedimensional discretization to the problem and ultimately develop a numerical scheme for which we prove a convergence result. 

(Brown) 
Global Stability of Solutions to a BetaPlane Equation  We study the motion of an incompressible, inviscid twodimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as $\beta$plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, nondegenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a “double null form” that annihilates interactions between waves with parallel frequencies and a Lemma for Fourier integral operators, which allows us to control a strong weighted norm and is based on a nondegeneracy property of the nonlinear phase function associated with the problem. Joint work with Fabio Pusateri; prior work with Tarek Elgindi. 

(UPenn) 
Minimal surfaces with bounded index  In this talk we will give a precise picture on how a sequence of minimal surfaces with bounded index might degenerate on given closed threemanifold. As an application, we prove several compactness results.  
(UMass Amherst) 
The Cauchy problem for wave maps on hyperbolic spaces  This talk is based on joint work with A. Lawrie and S.J. Oh on global existence and scattering for the small data Cauchy problem for wave maps on hyperbolic spaces $H^d$, $d\geq 4$. To motivate the problem I will begin by discussing previous results on the equivariant problem on H^2 where, in contrast to the Euclidean problem, there is a 1parameter family of finite energy harmonic maps. In the remainder of the talk I will present the gauge formulation of the wave maps equation, and describe new issues that arise in this context due to the geometry of the domain. I will then discuss Tao's caloric gauge and show how it can be used to resolve these geometric issues.  
(Zhejiang U) 
Long time existence for semilinear wave equations on asymptotically flat spacetimes  In this talk, we will talk about the long time existence of solutions to semilinear wave equations of the form $(\partial_t^2\Delta) u=u^p$, for small data with sufficient regularity and decay, on a large class of $(1+n)$dimensional Lorentzian nonstationary asymptotically flat backgrounds $(M, g)$. Under the assumption that uniform energy bounds and a weak form of local energy estimates hold forward in time, we obtain the sharp lower bounds of the lifespan for three dimensional subcritical and four dimensional critical cases. For the most delicate three dimensional critical case ($p=p_c$), we obtain the existence result up to $\exp(c \epsilon^{2(p1)})$, for many spacetimes including the nontrapping exterior domain, nontrapping asymptotically Euclidean space and Schwarzschild black hole spacetime.  
(ASU) 
Asymptotic rigidity of noncompact shrinking gradient Ricci solitons  Shrinking gradient Ricci solitons (shrinkers) are generalized fixed points of the Ricci flow equation and models for the geometry of a solution to the flow in the neighborhood of a developing singularity. There is mounting evidence to suggest that the possible asymptotic geometries of a complete noncompact shrinker are highly restricted, and all examples presently known are either locally reducible as products or smoothly asymptotic to regular cones at infinity. I will present some recent results obtained in part with Lu Wang in which we approach the uniqueness of complete noncompact shrinkers as a problem of parabolic unique continuation, and discuss their application to a conjectural classification in four dimensions.  
(Princeton) 
Smoothing conic KaehlerEinstein metrics  Conic Kaehler metrics have played a very important role in recent advances on complex geometry. In this expository talk. I will discuss some progresses on smoothing conic Kaehler metrics with Ricci curvature bounded from below, including conic KaehlerEinstein metrics. I will also discuss some applications.  
(La Sapienza) 
Blowingup solutions for Yamabetype problems  In this talk, I will discuss the existence and multiplicity of blowingup solutions for linear perturbation of Yamabe problem. I will also present some recent results concerning multiplicity of large conformal metrics with prescribed scalar curvature.  
(AT 3:00PM) 
(Rutgers) Monroe Martin Lecture 
Rigidity problems in several complex variables and complex geometry  In 1907, Poincare proved that any nonconstant holomorphic map sending a piece of the boundary of the unit ball in a complex space of dimension two into the boundary of the unit ball extends to an automorphism of the ball. This result fails for holomorphic functions of one variable and reveals a strong rigidity property for holomorphic functions of several variables. The program of Poincare that started in 1907 was later pursued further by Segre (1930's), Cartan (1940's) and ChernMoser (1974). The theory of SegreCartanChernMoser shows that real hypersyrfaces in a complex space of higher dimension posses many holomorphic invariants and holomorphic maps between hypersurfaces need to preserve those invariants and thus must be very rigid. Based on these invariants, Webster and the speaker established a several complex analysis version of the W. L. Chow theorem in 1977 (equidimensional case) and in 1994 (any codimensional case), respectively. These invariant and rigidity results have found many applications in the study of other rigidity problems in several complex variables, CR geometry, isolated complex singularity theory and complex geometry. The main goal of this talk is to give a survey on these lines of investigation in which the author has been involved in the past 20 years. The topics include: Gap rigidity for proper holomorphic maps between balls; rigidity problems for Milnor links and isolated normal complex singularities; rigidity of local holomorphic volume preserving maps between Hermitian symmetric spaces. 
(EPF Lausanne) 
On stability of type II blow up solutions for the critical nonlinear wave equation.  The talk will discuss a recent result showing that type II blow up solutions constructed by KriegerSchlagTataru are actually stable under small perturbations along a codimension one Lipschitz hypersurface in a suitable topology. This result is qualitatively optimal. Joint work with Stefano Burzio (EPFL).  
(Rutgers) 
NonK\"ahler Ricci flows and K\"ahler singularity models  We investigate Riemannian (nonK\"ahler) Ricci flow solutions that develop finitetime singularities with the property that parabolic rescalings at the singularities converge to singularity models that are shrinking K\"ahlerRicci solitons, specifically, the conjecturally stable ``blowdown soliton'' discovereed by Feldman, Ilmanen and Knopf. This is a joint work with Isenberg and Knopf.  
(EXTRA SEMINAR) 
(Bielefeld) 
Bilinear Restriction Estimates for General Phases  Bilinear restriction estimates were originally developed to make progress on the question of L^p estimates for the Fourier transform of compact hypersurfaces. However bilinear restriction estimates can also be stated in terms of products of transverse waves, and thus are closely connected to problems in dispersive PDE. In this talk we present new bilinear restriction estimates for general phases at multiple scales, which extend recent estimates of Bejenaru, LeeVargas, and Tao (among others). These estimates can be extended to hold in the framework of adapted function spaces, and hence have applications to dispersive PDE in scale invariant settings. In particular, in joint work with Sebastian Herr, we give an application to scattering for the DiracKleinGordon system. 
Archive of Analysis Seminar:
Fall 2016 Spring 2016 Fall 2015 Spring 2015 Fall 2014 Spring 2014 Fall 2013