Columbia Analysis Seminar

Here are our upcoming seminars:

• January 26, 2024: Conghan Dong (Stony Brook University), Stability of Euclidean 3-space for the positive mass theorem

  The Positive Mass Theorem of R. Schoen and S.-T. Yau in dimension 3 states that if $(M^3, g)$ is asymptotically flat and has nonnegative scalar curvature, then its ADM mass $m(g)$ satisfies $m(g) \geq 0$, and equality holds only when $(M, g)$ is the flat Euclidean 3-space $\mathbb{R}^3$. We show that $\mathbb{R}^3$ is stable in the following sense. Let $(M^3_i, g_i)$ be a sequence of asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that $m(g_i)$ converges to 0. Then for all $i$, there is a domain $Z_i$ in $M_i$ such that the area of the boundary $\partial Z_i$ converges to zero and the sequence $(M_i \setminus Z_i , \hat{d}_{g_i} , p_i )$, with induced length metric $\hat{d}_{g_i}$ and any base point $p_i \in M_i \setminus Z_i$, converges to $\mathbb{R}^3$ in the pointed measured Gromov-Hausdorff topology. This confirms a conjecture of G. Huisken and T. Ilmanen. This talk is based on joint work with Antoine Song.
  
  

• February 2, 2024, (NOTICE THE TIME and ROOM CHANGE for the two speakers):
  
  
  •    from 11:30am to 12:30pm in room 312: Chao-Ming Lin (Ohio State University), On the solvability of general inverse $\sigma_k$ equations

    In this talk, first, I will introduce general inverse $\sigma_k$ equations in Kähler geometry. Some classical examples are the complex Monge–Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation. Second, by introducing some new real algebraic geometry techniques, we can consider more complicated general inverse $\sigma_k$ equations. Last, analytically, we study the solvability of these complicated general inverse $\sigma_k$ equations.
    
    

  •    from 2:30pm to 3:30pm in room 507: Lili He (Princeton University), The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

    I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.
    
    

• February 9, 2024: Ruobing Zhang (Princeton University), Collapsing Einstein manifolds with special holonomy

  We will talk about recent developments in the metric geometry of collapsing Einstein manifolds. We will particularly focus on the Calabi-Yau and hyperkähler cases.
  
  

• February 16, 2024: Yakov Shlapentokh-Rothman (University of Toronto), Polynomial Decay for the Klein-Gordon Equation on the Schwarzschild Black Hole

  We will start with a review of previous instability results concerning solutions to the Klein-Gordon equation on rotating Kerr black holes and the corresponding conjectural consequences for the dynamics of the Einstein-Klein-Gordon system. Then we will discuss recent work where we show that, despite the presence of stably trapped timelike geodesics on Schwarzschild, solutions to the corresponding Klein-Gordon equation arising from strongly localized initial data nevertheless decay polynomially. Time permitting we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums. The talk is based on joint work(s) with Federico Pasqualotto and Maxime Van de Moortel.
  
  

• March 1, 2024: Hans Ringstrom (KTH), Proving curvature blow up in cosmology

  In a recent joint work with Hans Oude Groeniger and Oliver Petersen, we identify a general condition on initial data ensuring big bang formation, including curvature blow up. The purpose of the talk is to describe some of the key arguments needed to obtain this result. In particular, I describe the gauge choice, the substitute for a background solution, the bootstrap assumptions, and give a rough idea of the energy estimates needed to close the bootstrap argument.
  
  

• March 8, 2024, (NOTICE THE TIME and ROOM CHANGE for the two speakers):
  
  
  •    from 11:30am to 12:30pm in room 312: Renato Velozo Ruiz (University of Toronto), On linear and non-linear stability of collisionless systems on black hole exteriors

    I will present upcoming linear and non-linear stability results for collisionless systems on spherically symmetric black holes. On the one hand, I will discuss the decay properties of massive Vlasov fields on the exterior of Schwarzschild spacetime. On the other hand, I will discuss an asymptotic stability result for the exterior of Schwarzschild as a solution to the Einstein-massless Vlasov system, assuming spherical symmetry. These results follow via concentration estimates on suitable stable manifolds in phase space.
    
    

  •    from 2:15pm to 3:15pm in room 417: Pei-Ken Hung (University of Illinois Urbana-Champagne), Higher-order asymptotics for a class of non-linear evolution equations

    Understanding solutions near their singularities is a fundamental topic in PDE. The pioneering works of Leon Simon established the uniqueness of blow-ups for a broad class of geometric PDEs. Subsequently, the investigation of higher-order behavior becomes a crucial step for further analysis. In this talk, we will provide a complete description of higher-order asymptotics based on analytic gradient flows. As a consequence, we verify Thom's gradient conjecture in the context of geometric PDEs. This talk is based on joint work with B. Choi.
    
    

• March 22, 2024: Claude Warnick (Cambridge University), (In)stability of quasinormal frequencies of black holes

  A perturbed black hole rings down by producing radiation at certain fixed (complex) frequencies - the quasinormal frequencies. These frequencies can be identified with the spectrum of a non-self adjoint operator derived from the evolution equation of the particular field of interest. Thanks to accurate measurements of gravitational waves, the quasinormal spectrum of a black hole is increasingly an observable quantity. A natural question is whether the quasinormal spectrum is stable to small perturbations of the underlying black hole spacetime. I will explain how this question is closely connected to the non-standard nature of the underlying spectral problem, and present explicit results that shed light on the problem.
  
  

• April 5, 2024 (ON ZOOM): Zhenhua Liu (Princeton University), General behavior of area-minimizing subvarieties

  We will review some recent progress on the general geometric behavior of homologically area-minimizing subvarieties, namely, objects that minimize area with respect to homologous competitors. They are prevalent in geometry, for instance, as holomorphic subvarieties of a Kahler manifold, or as special Lagrangians on a Calabi-Yau, etc. A fine understanding of the geometric structure of homological area-minimizers can give far-reaching consequences for related problems. Camillo De Lellis and his collaborators have proven that area-minimizing integral currents have codimension two rectifiable singular sets. A pressing next question is what one can say about the geometric behavior of area-minimizing currents beyond this. Almost all known examples and results point towards that area-minimizing subvarieties are subanalytic, generically smooth, calibrated, and have a priori curvature bounds. It is natural to ask if these hold in general. In this direction, we prove that all of these properties thought to be true generally and proven to be true in special cases are totally false in general. We prove that area-minimizing subvarieties can have fractal singular sets. Smoothable singularities are non-generic. Calibrated area minimizers are non-generic. A priori L^2 curvature bounds even fail for holomorphic subvarieties. Consequently, we answer several conjectures of Frederick J. Almgren Jr., Frank Morgan, and Brian White from the 1980s.
  
  

• April 19, 2024: Francisco Martín (Universidad de Granada), Semi-graphical Translators of the Mean Curvature Flow

  A soliton is a special solution to a partial differential equation that maintains its shape and moves at a constant velocity. In the context of mean curvature flow, a translating soliton is a solution to the mean curvature flow equation that moves by a constant velocity in the direction of a vector. Translating solitons are particularly interesting because they provide insights into the behavior of evolving surfaces. On the other hand, we say that a surface is semi-graphical if when we remove a discrete set of vertical lines, then the resulting surface is the graph of a smooth function. We are going to provide a classification of all the semi-graphical translator in Euclidean 3-space. First, we will describe a comprehensive zoo of all examples of this type of translators, and then we will focus on classification arguments. We will conclude with some open problems. This talk summarizes various joint works with D. Hoffman and B. White, on one hand, and with M. Saez and R. Tsiamis, on the other.
  
  

• April 26, 2024: Hui Yu (National University of Singapore), Rigidity of global solutions to the thin obstacle problem

  The thin obstacle problem is a free boundary problem concerning the shape of an elastic membrane resting on a lower-dimensional obstacle. In this talk, we discuss some rigidity properties of solutions in the entire space. We see very rigid behaviors when the solution grows quadratically at infinity. When the solution has higher rate of growth, we see no rigidity. This talk is based on joint works with Simon Eberle (BCAM) and Xavier Fernandez-Real (EPFL).
  
  

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Here are our past seminars in Fall 2023

• September 15, 2023: Stefano Vita (University of Turin), Boundary Harnack principle on nodal domains

  Given a uniformly elliptic equation in divergence form, let us consider two solutions $u,v$ which share their zero sets $Z(u)\subseteq Z(v)$. Then, regularity features of the ratio $w=v/u$ across the nodal set of $u$ is equivalent to Schauder estimates for continuous solutions of some elliptic equations having coefficients which degenerate as $u^2$ on $Z(u)$.
  
  

• September 22, 2023 from 10:30am to 11:30am (NOTICE THE TIME CHANGE): John Anderson (Stanford University), Formation of shocks for the Einstein-Euler system

  In this talk, I hope to describe elements of proving stable shock formation for the Einstein-Euler system in the setting of potential flow. This involves proving that the fluid variables blow up in a specific way while the gravitational metric remains comparatively smooth. I'll first describe where this fits into the study of shocks, and why it is appropriate to call this singularity formation result a shock formation result. Then, I will go through some of the most important parts of Christodoulou's landmark proof of shock formation for potential flow on a fixed background, as well as followup breakthrough works by Luk-Speck allowing for vorticity. This will show the main difficulty present in proving that the gravitational metric remains comparatively smooth in the case of Einstien-Euler. It essentially arises from the fact that the speed of sound is less than the speed of light. In the remaining time, I will try to give the main idea in overcoming this difficulty. This is work in progress with Jonathan Luk.
  
  

• September 29, 2023: Chen-Chih Lai (Columbia University), Thermal effects in bubble oscillations

  We study the thermal decay of bubble oscillation in an incompressible liquid. In this talk, we consider two models, both systems of nonlinear PDEs with a moving boundary: the complete mathematical formulation (full model) and an approximate model proposed by A. Prosperetti in [J. Fluid Mech. 1991]. These two models share a one-parameter manifold of spherical equilibria, parametrized by the bubble mass. Within the approximate model, we prove that the manifold of spherical equilibria is an attracting centre manifold against small spherically symmetric perturbations and that solutions approach this manifold at an exponential rate as time advances. We also examine the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic external sound field. We prove that this periodically forced system admits a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations. Finally, we report on results regarding the characterization of all equilibria within each model. For the approximate system, we prove that all equilibrium bubbles are spherically symmetric through an application of Alexandrov’s theorem on closed constant-mean-curvature surfaces. Furthermore, the aforementioned family of spherical equilibria encompasses all the spherical equilibria of the approximate system. However, within the full model, this family is embedded in a larger family of spherically symmetric solutions. If time permits, I will discuss a work in progress on asymmetric dynamics of these models and future directions. This talk is based on joint work with Michael I. Weinstein (arXiv:2207.04079, arXiv:2305.03569, and work in progress).
  
  

• October 6, 2023: Ravi Shankar (Princeton University), A doubling approach for two sigma-k PDEs

  The interior regularity for viscosity solutions of the sigma-2 equation is the remaining case of the Monge-Ampere sigma-k family to be understood. The dimension two case was done in the 1950’s by Heinz, and the dimension three case was done in 2008 by Warren and Yuan. In joint works with Yu Yuan, we show regularity in dimension four for sigma-2. Our argument uses a doubling inequality for the Hessian to propagate Alexandrov-Savin partial regularity. A similar argument gives a proof of interior regularity for strictly convex solutions of the Monge-Ampere (sigma-n) equation in all dimensions, using a geometric approach distinct from Pogorelov’s maximum principle of the 70’s.
  
  

• October 27, 2023: Tristan Ozuch (MIT), Selfduality and instabilities of Einstein metrics and Ricci solitons

  Einstein metrics and Ricci solitons are the fixed points of Ricci flow and model the singularities forming. They are also critical points of natural functionals in physics. Their stability in both contexts is a crucial question, since one should be able to perturb away from unstable models. I will present new and upcoming results about the stability of these metrics in dimension four in joint works with Olivier Biquard and with Keaton Naff. They rely on a specificity of dimension four called selfduality.
  
  

• November 3, 2023: Ryan Unger (Princeton University), Retiring the third law of black hole thermodynamics

  In this talk, I will present a construction of regular initial data for the Einstein-Maxwell-charged scalar field system collapsing to extremal Reissner-Nordström black holes in finite time. In particular, our result can be viewed as a definitive disproof of “the third law of black hole thermodynamics.” This has opened the door to studying a new type of critical phenomenon on the black hole formation threshold which we call “extremal critical collapse.” This is joint work with Christoph Kehle (ETH Zurich).
  
  

• November 10, 2023 from 11:10am to 12:10pm (NOTICE THE TIME CHANGE): Donatella Danielli (Arizona State University), Obstacle problems for fractional powers of the Laplacian

  In this talk we will discuss a sampler of obstacle-type problems associated with the fractional Laplacian $(−\Delta)^s . Our goals are to establish regularity properties of the solution and to describe the structure of the free boundary. To this end, we combine classical techniques from potential theory and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas.
  
  

• November 17, 2023: Demetre Kazaras (Michigan State University), Almost rigid Riemannian manifolds and drawstrings

  The topic of this talk is the structure of Riemannian 3-manifolds satisfying scalar curvature lower bounds. An influential theorem of Schoen-Yau and Gromov-Lawson says that the only Riemannian metrics of nonnegative scalar curvature on the n-torus are flat. A program of Sormani poses the associated 'almost rigidity problem', asking us to describe a manner in which Riemannian 3-tori of almost nonnegative scalar curvature are close to flat tori. I will describe a new 'drawstring' phenomenon in this problem, present some negative results, as well as some positive progress on almost rigidity problems related to Llarull's theorem and the positive mass theorem.
  
  

• December 1, 2023: Allen Fang (Munster University), Wave behavior in the vanishing cosmological constant limit

  Black hole stability is a central topic in mathematical relativity that has seen numerous advancements in recent years. Both the Kerr-de Sitter and the Kerr black hole spacetimes have been proven to be stable in the slowly-rotating regime. However, the methods used have been markedly different, as well as the decay rates proven. Perturbations of Kerr-de Sitter converge exponentially back to a nearby Kerr-de Sitter black hole, while perturbations of Kerr only converge polynomially back to the family. In this talk, I will speak about wave behavior that is uniform in the cosmological constant by considering solutions to the model Regge-Wheeler equations in Kerr(-de Sitter). The main point is a careful handling of the relevant estimates on the region of the spacetime far from the black hole. This provides a first step into understanding the uniform (in the cosmological constant) stability of black hole spacetimes. This is joint work with Jeremie Szeftel and Arthur Touati.
  
  

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The organizers,

Daniela De Silvia, Elena Giorgi