(Princeton) 
The problem of global regularity for water waves.  The water waves equations are a system of evolution equations modeling the motion of waves, like those in the surface of the ocean. After introducing the system we will discuss some of the works done in recent years on the question of longtime regularity. We will then present a result, joint with Deng, Ionescu and Pausader, about global existence of smooth solutions for the 3d gravitycapillary water waves system in infinite depth. The main difficulties in this problem are the slow decay of linear solutions and the presence of large sets of resonant interactions.  
(JHU) 
Concentration Compactness for the Critical MaxwellKleinGordon Equation  The MaxwellKleinGordon Equation is a system of evolution equations that describes the interaction of an electromagnetic field with a charged particle field. In this talk we discuss a proof of global regularity, scattering and a priori bounds for solutions to the energy critical MaxwellKleinGordon equation relative to the Coulomb gauge for essentially arbitrary smooth data of finite energy. The proof is based upon a novel "twisted" BahouriGérard type profile decomposition and a concentration compactness/rigidity argument by KenigMerle, following the method developed by KriegerSchlag in the context of energy critical wave maps. This is joint work with Joachim Krieger.  
(McGill) 
Classical Neumann problems for Hessian equations and geometric applications  The classic Neumann problem for laplace equation has many geometric applications. For example, Reilly used its solution to give a new proof of Minkowski inequality. Recently, Xinan Ma and Guohuan Qiu, have proved the existence of the Neumann problems for Hessian equations in uniformly convex domain in Rn. Motivated from Reilly and MaQiu's work, Chao Xia and I also find geometric applications about classical Neumann problems for Hessian equations. We will talk about how to prove the existence of classical Neumann problems under the uniformly convex domain. Then we use the solution of the classical Neumann problem to give a new proof of a family of AlexandrovFenchel inequalities arising from convex geometry.  
(Stony Brook) 
Regularity and Degeneration of Einstein Metrics  In this talk, we will review some classical and new results in the regularity and degeneration theory of Einstein manifolds. That is, we will discuss the effective curvature estimates on a Einstein manifold and the limiting behavior for a sequence of Einstein manifolds. A reasonable starting point is the regularity theory of noncollapsing Einstein manifolds, which has been well established in recent years. However, the regularity and degeneration behavior of collapsing Einstein manifolds are much harder to handle. Without any extra assumption, there are few analytic tools in the general collapsed context. In this direction, we will introduce some new regularity theorems (joint work with Aaron Naber) and open questions.  
(MSU) 
A minmax formula for Lipschitz operators that satisfy the global comparison principle  We investigate Lipschitz maps, I, mapping $C^2(D) \to C(D)$, where $D$ is an appropriate domain. The global comparison principle (GCP) simply states that whenever two functions are ordered in D and touch at a point, i.e. $u(x)\leq v(x)$ for all $x$ and $u(z)=v(z)$ for some $z \in D$, then also the mapping I has the same order, i.e. $I(u,z)\leq I(v,z)$. It has been known since the 1960’s, by Courr\`{e}ge, that if I is a linear mapping with the GCP, then I must be represented as a linear driftjumpdiffusion operator that may have both local and integrodifferential parts. It has also long been known and utilized that when I is both local and Lipschitz it will be a minmin over linear and local driftdiffusion operators, with zero nonlocal part. In this talk we discuss some recent work that bridges the gap between these situations to cover the nonlinear and nonlocal setting for the map, I. These results open up the possibility to study DirichlettoNeumann mappings for fully nonlinear equations as integrodifferential operators on the boundary. This is joint work with Nestor Guillen  
(Harvard U) 
Entropy and selfshrinkers of the mean curvature flow  The ColdingMinicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropystable selfshrinkers that may have a small singular set.  
(MIT) 
A Proof of Onsager’s Conjecture  In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured that weak solutions to the incompressible Euler equations may violate the law of conservation of energy if their spatial regularity is below 1/3Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3\epsilon)Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Höldercontinuous Euler flows that fail to conserve energy, and was later improved by Isett and by BuckmasterDe LellisSzékelyhidi to obtain further partial results towards Onsager's conjecture. The proof to be discussed of the full conjecture combines a new ingredient in the convex integration scheme due to DaneriSzékelyhidi with a new "gluing approximation" technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.  
(Princeton U) 
Scattering theory and blueshift instabilities on Kerr exterior and interior spacetimes.  In the first part of the talk, which concerns joint work with Dafermos and Rodnianski, we will present results corresponding to the "existence and uniqueness of scattering states" and "asymptotic completeness" for the wave equation on Kerr exterior spacetimes. These results are stated with respect to function spaces which allow for the solutions to exhibit certain singular behaviors along the event horizon. In the second part of the talk, which concerns joint work with Dafermos, we will show that it is in fact necessary to allow such singular behavior, and furthermore, exploiting essentially the same mechanism, we will also show that solutions to the wave equation generically have a singular behavior along the CauchyHorizon in Kerr black hole interiors.  
(Amherst) 
Pointwise estimates for Generated Jacobian Equations  Generated Jacobian Equations (GJE) are a class of nonlinear, degenerate elliptic equations introduced by Trudinger, motivated by the analysis of reflecting surfaces in geometric optics. I will present a new regularity result for weak solutions of these equations, building in particular on work of Caffarelli for the real MongeAmpère equation and work of Figalli, Kim, and McCann in optimal transport. An example of an equation covered by the theory is the point source nearfield reflector problem, which already shows markedly different regularity phenomena when compared to optimal transport, as shown by the analysis and examples by Karakhanyan and Wang. A central tool in our approach is the derivation of certain pointwise inequalities connected to the BlaschkeSantaló inequality. This talk is based on joint works with Jun Kitagawa.  
(MIT) 
The freeboundary Brakke flow  A surface has geometric freeboundary in a barrier hypersurface if its boundary meets the barrier orthogonally, like a bubble on a bathtub. We extend Brakke's weak notion of mean curvature flow to have a freeboundary condition, and using toy examples we show why this extension is necessary. Contrary to the classical flow, for which the barrier is ``invisible,'' our notion allows for the flow to ``pop'' or ``break up'' upon tangential contact with the barrier.  


(NYU) 
Quantitative connectivity, (weighted) Poincare inequalities and differentiability in doubling metric measure spaces.  Analysis on metric spaces concerns the study of first order analysis in nonsmooth spaces, and their relations to the underlying geometry. This talk will focus on three related notions: notions of connectivity, Poincare inequalities and differentiability. I will outline how recent developments by myself and others have closely linked these seemingly unrelated concepts to each other. I will present two main theorems from my paper and the ideas involved in proving them. The first concerns the characterization of Poincare inequalities on doubling metric measure spaces in terms of a simple quantitative connectivity condition. The second theorem partially resolves an open question of Cheeger and Kleiner on the local geometry of Lipschitz differentiability spaces, and proves a new type of rectifiability result for a subclass of them. Finally we relate the new concepts to Muckenhoupt weights and new types of selfimprovement results in the spirit of KeithZhong are presented." 
Archive of Analysis Seminar:
Spring 2016 Fall 2015 Spring 2015 Fall 2014 Spring 2014 Fall 2013