Concluding Conference on Nonlinear Equations. Conference Program. 9:00-10:00am Camillo De Lellis, Universität Zürich

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1 Concluding Conference on Nonlinear Equations Conference Program Friday, April 8 9:00-10:00am Camillo De Lellis, Universität Zürich Title: A Nash-Kuiper theorem for $C^{1,1/5}$ isometric immersions of disks Abstract: In a joint work with Dominik Inauen and L\'aszl\'o Sz\'ekelyhidi we prove that, given a $C^2$ Riemannian metric $g$ on the $2$-dimensional disk, any short $C^1$ embedding can be uniformly approximated with $C^{1,\alpha}$ isometric embeddings for any $\alpha < \frac{1}{5}$. The same statement with $C^1$ isometric embeddings is a groundbreaking result due to Nash and Kuiper. The previous Hoelder threshold, 1/7, was first announced in the sixties by Borisov. If time allows I will also discuss the connection with a conjecture of Onsager on weak solutions to the Euler equations. 10:00-10:15am Break 10:15-11:15am Xu-Jia Wang, Australian National University Title: Monge s mass transport problem Abstract: The optimal transportation problem can be formulated as a Monge-Ampere type equation, and the existence and regularity of optimal mappings have been established under certain conditions. Monge s original problem is one of the most interesting cases and is at the borderline of these conditions. With my collaborators Qi-Rui Li and Filippo Santambrogio, we recently studied the regularity of Monge s problem and observed some delicate results. Namely we proved that in a smooth approximation, the eigenvalues of the Jacobian matrix of the optimal mapping are uniformly bounded but the mapping itself may not be Lipschitz continuous. But in dimension two the mapping is continuous. In this talk I will discuss recent development in this direction. 11:30am-12:30pm Peng-Fei Guan, McGill University Title: "The Weyl isometric embedding problem in general $3$-d Riemannian manifolds Abstract: The classical problem concerns the isometric embedding of positively curved surfaces $(\mathbb S^2, g)$ to $\mathbb R^3$. The problem was solved in 1950s by Nirenberg. The similar problem was also considered by Pogorelov for hyperbolic space $\mathbb H^3$. The Weyl problem plays important role in the definition of quasilocal masses in general relativity, in particular, the work of Brown-York, Shi-Tam, Liu-Yau and Wang-Yau. For general ambient space (which may not be a space form), there are some works on the curvatures estimates (Guan-Lu) and openness (Li-Wang) for immersed surfaces. In this talk, we will concentrate on recent work of Siyuan Lu on localized curvature estimate. It is a generalization of Heinz's result, but using more sophisticated estimates of Shi-Tam and Wang-Yau on quasilocal masses. This estimate yields $C^0$ estimate of the embeddings, resolving an open problem after the works of Guan-Lu and Li-Wang. 1

2 12:30-2:00pm Break for Lunch 2:00-3:00pm Blake Temple, UC Davis Title: "An instability in the Standard Model of Cosmology Creates the Anomalous Acceleration without Dark Energy Abstract: We introduce a new asymptotic ansatz for smooth, spherical perturbations of the Standard Model of Cosmology (SM) which applies during the p=0 epoch, and prove that such perturbations trigger an instability in the SM on the scale of the supernova data. The instability creates a large, central region of uniform under-density which expands faster than the SM, and this accelerated uniform expansion introduces into the SM precisely the same range of corrections to redshift vs luminosity as are produced by the cosmological constant in the theory of Dark Energy. A phase portrait of the instability places the Standard Model (SM) at a classic unstable saddle rest point, and universality is exhibited in the sense that all sufficiently small perturbations evolve to a nearby stable rest point corresponding to Minkowski space. We then prove that this instability is triggered by a one parameter family of self-similar waves from the radiation epoch p = (c! /3)ρ when the pressure drops to p = 0. The authors previously proposed this family as possible time-asymptotic wave patterns for perturbations of the SM at the end of the radiation epoch. Using numerical simulations, we calculate the unique wave in the family that accounts for the same values of the Hubble constant and quadratic correction to redshift vs luminosity as in a universe with seventy percent Dark Energy, Ω_Λ 7. A numerical simulation of the third order correction associated with that unique wave establishes a testable prediction that distinguishes this theory from the theory of Dark Energy. This explanation for the anomalous acceleration, based on instabilities in the SM together with simple wave perturbations from the radiation epoch that trigger them, provides perhaps the simplest mathematical explanation for the anomalous acceleration of the galaxies that does not invoke Dark Energy. Joint work with Joel Smoller and Zeke Vogler. 3:00-3:15pm Break 3:15-4:15pm Lydia Bieri, University of Michigan Title: The Einstein Equations and Gravitational Radiation Abstract: In Mathematical General Relativity (GR) the Einstein equations describe the laws of the universe. This system of hyperbolic nonlinear pde has served as a playground for all kinds of new problems and methods in pde analysis and geometry. A major goal in the study of these equations is to investigate the analytic properties and geometries of the solution spacetimes. In partic- ular, fluctuations of the curvature of the spacetime, known as gravitational waves, have been a highly active research topic. A few weeks ago, it was confirmed that advanced LIGO detected gravitational waves. Understand- ing gravitational radiation is tightly interwoven with the study of the Cauchy problem in GR. I will talk about geometric-analytic results on gravitational radiation and the memory effect of gravitational waves. We will connect the mathematical findings to experiments. I will also address recent work with David Garfinkle on gravitational radiation in asymptotically flat as well as cosmological spacetimes. 4:15-4:30pm Break 2

3 4:30-5:30pm Valentino Tosatti, Northwestern University Title: Adiabatic limits of Ricci-flat Kahler metrics Abstract: I will discuss the behavior of Ricci-flat Kahler metrics on the total space of a fibration in the adiabatic limit when the volume of the fibers shrinks to zero. On the geometric side, this problem is relevant for the Strominger-Yau-Zaslow picture of mirror symmetry. On the analytic side, it is equivalent to studying solutions of a degenerating family of complex Monge-Ampere equations. Saturday, April 9 9:00-10:00am Duong H. Phong, Columbia University Title: On Strominger systems and Fu-Yau equations Abstract: Strominger systems are a system of equations arising in string theory, which are potentially quite important for complex geometry as an analogue of the curvature conditions for canonical metrics in a non-k\ ahler setting. The first non-perturbative solutions of Strominger systems were obtained by J.X. Fu and S.T. Yau some 10 years ago. These solutions have now led to many new equations, of Hessian and Monge-Ampere type, both elliptic and parabolic, which are also interesting from the point of view of non-linear partial differential equations. We discuss these developments, with particular emphasis on a priori estimates. This is joint work with S. Picard and X. Zhang. 10:15-11:15am Slawomir Kolodziej, Jagiellonian University Title: Stability of weak solutions of the complex Monge-Ampère equation on compact Hermitian manifolds Abstract: This is joint work with Nguyen Ngoc Cuong. Let (X, ω) be a compact Hermitian manifold of complex dimension n. We study the weak solutions to the complex Monge- Ampère equation (ω+dd c φ) n = fω n, ω+dd c φ 0 where 0 f L p (X,ω n ), p > 1, and dd c = i/π, with the inequality understood in the sense of currents. For strictly positive right hand side a stability statement is given. It is used to show Hölder continuity of solutions and an extension of Székelyhidi - Tosatti theorem from Kähler to Hermitian manifolds. 10:30-10:45pm Break 11:30am-12:30pm Luis Caffarelli, UT Austin Title: "Non local minimal surfaces and their interactions Abstract: In this lecture I will review aspects of non local minimal surfaces Movement by non local mean curvature arises when considering heat flows for fractional Laplacians and resemble neighbor to neighbor interaction but at a large scale. 3

4 We will discuss the properties of the corresponding stationary problem, and their interaction with bulk diffusion and among themselves 2:00-3:00pm Mihalis Dafermos, Princeton University Title: "The interior of dynamical vacuum black holes and the strong cosmic censorship conjecture in general relativity Abstract: I will discuss recent work on the structure of black hole interiors for dynamical vacuum spacetimes (without any symmetry) and what this means for the question of the nature of generic singularities in general relativity and the celebrated strong cosmic censorship of Penrose. This is joint work with Jonathan Luk. 12:15-2:00pm Break for lunch 3:15-4:15pm Mu-Tao Wang, Columbia University Title: The stability of Lagrangian curvature flows Abstract: I shall present some new stability theorems of Lagrangian curvature flows in both the cotangent bundle case and the Calabi-Yau case. The talk will be based on joint work with Knut Smoczyk and Mao-Pei Tsui, and joint work with Chung-Jun Tsai. 4:30-5:30pm Chiu-Chu Melissa Liu, Columbia University Title: Counting curves in a quintic threefold Abstract: A quintic threefold is a degree 5 hypersurface in the 4-dimensional complex projective space. By the Calabi conjecture proved by Yau, a smooth quintic threefold admits Ricci flat Kahler metrics. In this talk, I will survey conjectures and results on counting holomorphic curves in a smooth quintic threefold in the past 25 years, including recent joint work of Huai-Liang Chang, Jun Li, Wei-Ping Li, and myself. Sunday, April 10 9:00-9:45am Richard Schoen, Stanford University Title: Metrics of fixed area on high genus surfaces with largest first eigenvalue Abstract: We will describe the problem of maximizing the first eigenvalue on a closed surface among metrics of a fixed area. This is a nonlocal geometric variational problem whose solutions arise as the induced metrics on certain minimal immersions of the surface into a sphere. We will explain a very general existence theorem and describe questions related to the geometry of the solutions. 10:00-10:15am Break 4

5 10:15-11:15am Clifford Taubes, Harvard University Title: The zero loci of Z/2 harmonic spinors in dimensions 2, 3 and 4 Abstract: Supposing that X is a Riemannian manifold, a Z/2 spinor on X is defined by a data set consisting of a closed set in X to be denoted by Z, a real line bundle over X Z, and a nowhere zero section on X Z of the tensor product of the real line bundle and a spinor bundle. The set Z and the spinor are jointly constrained by the following requirement: The norm of the spinor must extend across Z as a continuous function vanishing on Z. In particular, the vanishing locus of the norm of the spinor is the complement of the set where the real line bundle is defined, and hence where the spinor is defined. The Z/2 spinor is said to be harmonic when it obeys a first order Dirac equation on X Z. This lecture will describe the structure of the set Z for a Z/2 harmonic spinor on a manifold of dimension either two, three or four. 11:30am-12:30pm Tai-Peng Tsai, University of British Columbia Title: "Forward Self-Similar and Discretely Self-Similar Solutions of the 3D incompressible Navier-Stokes Equations Abstract: For 3D incompressible Navier-Stokes Equations in the whole space, the existence of forward self-similar solutions with large data was only shown recently by Jia and Sverak based on Leray Schauder theorem and a priori bounds for local Leray solutions near initial time. Their approach was adapted by myself to construct discretely self-similar (DSS) solutions either with DSS factor sufficiently close to 1 or for axisymmetric data. These solutions are necessarily regular. They are of interest since they may not be unique, corresponding to stationary and Hopf bifurcations for associated Leray equations. In this talk I will report two results: The first is a joint work with Mikhail Korobkov on the construction of self-similar solutions in the half space. In the half space the local Leray solution theory is not known, and we get a priori bound using Leray's contradiction argument and reduce the problem to the study of Euler equations in the half space. The second result is a joint work with Zachary Bradshaw on the construction of DSS solutions for general data. It is a weak solution theory, based on a new explicit a priori bound, and the initial data is only required to be DSS and locally L^3. Such solutions may not be regular and are good candidates for the failure of eventual regularity. Note that both results give new constructions of self-similar solutions in the whole space, and the latter one trivializes the existence problem. 5