Columbia Analysis Seminar

Here are our upcoming seminars:

• September 13, 2024: Warren Li (Princeton University), BKL bounces outside homogeneity

  In the latter half of the 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general ansatz for solutions to the Einstein equations possessing a (spacelike) singularity. They suggest that, near the singularity, the evolution of the spacetime geometry at different spatial points decouples and is well-approximated by a system of autonomous nonlinear ODEs, and further that general orbits of these ODEs resemble a (chaotic) cascade of heteroclinic orbits called "BKL bounces". In this talk, we present recent work verifying the validity of BKL's heuristics in a large class of symmetric, but spatially inhomogeneous, spacetimes which exhibit (up to one) BKL bounce on causal curves reaching the singularity. In particular, we prove AVTD behavior (i.e. decoupling) even in the presence of inhomogeneous BKL bounces. The proof uses nonlinear ODE analysis coupled to hyperbolic energy estimates, and one hopes our methods may be applied more generally.
  
  

• September 20, 2024: Ovidiu-Neculai Avadenei (UC Berkeley), Low regularity well-posedness for the Generalized Surface Quasi-Geostrophic front equation

  We consider the well-posedness of the generalized surface quasi-geostrophic (gSQG) front equation. By making use of the null structure of the equation, we carry out a paradifferential normal form analysis in order to obtain balanced energy estimates, which allows us to prove the local well-posedness of the g-SQG front equation in the non-periodic case at a low level of regularity (in the SQG case, this is only one half of a derivative above scaling). In addition, we establish global well-posedness for small and localized rough initial data, as well as modified scattering, by using the testing by wave packet approach of Ifrim-Tataru. This is joint work with Albert Ai.
  
  

• September 27, 2024: Istvan Kadar (Princeton University), Construction of multi-soliton solutions for semilinear equations in dimension 3

  The existence of multi black hole solutions in asymptotically flat spacetimes is one of the expectation from the final state conjecture. In this talk, I will present preliminary works in this direction via a semilinear toy model in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.
  
  

• October 4, 2024: Nestor Guillen (Texas State University), The Landau equation does not blow up

  The Landau equation is a fundamental equation in kinetic theory -- the "correction" to Boltzmann's equation for particles interacting through a Coulomb potential. For the space homogeneous equation it can be described as a nonlocal, quasilinear heat equation with a quadratic nonlinearity. This last feature led to the initial expectation of finite time-blow up for solutions of the equation. Proving finite time blow up or global existence for this equation has been a chief concern in the analysis of kinetic PDE for the last couple of decades. In work with Luis Silvestre we show the monotonicity of the Fisher information for smooth solutions of the homogeneous Landau equation, a fact that guarantees global in time existence of smooth solutions. This is proved for a broad range of collision potentials including Coulomb. This result is made possible by three ingredients: 1) a new “lifting procedure” of the Landau equation involving a linear degenerate parabolic PDE in double the number of variables, 2) a decomposition of this PDE in layered-rotations and 3) a functional inequality closely related to the log-Sobolev inequality on the sphere. In this talk I will describe all three ingredients, starting by motivating some of them for the simpler case of the heat equation
  
  

• October 18, 2024: Connor Mooney (UC Irvine), The Lawson-Osserman conjecture for the minimal surface system

  In their seminal work on the minimal surface system, Lawson and Osserman conjectured that Lipschitz graphs that are critical points of the area functional with respect to outer variations are also critical with respect to domain variations. We will discuss the proof of this conjecture for two-dimensional graphs of arbitrary codimension. This is joint work with J. Hirsch and R. Tione.
  
  

• October 25, 2024: Hamed Masaood (Princeton University), A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole

  I will talk about the scattering problem in general relativity, and present a construction of a scattering theory resolving the problem for the linearised Einstein equations in a double null gauge against a Schwarzschild background. This is done by first constructing a scattering theory for the gauge invariant components of the linearised system via the spin $\pm2$ Teukolsky equations, and this is the subject of the first part of the talk. I will then discuss how this theory can extended to the full system. Key to this step is the identification of suitable asymptotic gauge conditions on scattering data. Here, a Bondi-adapted double null gauge is shown to provide the necessary gauge rigidity, in a manner that enables the identification of Hilbert-space isomorphism between finite energy scattering data and a suitable space of finite energy Cauchy data. In particular, the gauge conditions made on scattering data will allow for a global treatment of the BMS group, and the memory effect at past and future null infinity.
  
  

• November 1, 2024: Federico Franceschini (IAS), The dimension and behaviour of singularities of stable solutions to semilinear elliptic equations

  Let $f(t)$ be a convex, positive, increasing nonlinearity. It is known that stable solutions of $-\Delta u =f(u)$ can be singular (i.e., unbounded) if the dimension $n \ge 10$. Brezis conjectured that if $x=0$ is such a singular point, then $f'(u(x))$ blows-up like $|x|^{2-n}$. Villegas showed that such a strong statement fails for general nonlinearities. In this talk, we prove — for all nonlinearities — a version of Brezis conjecture, which is essentially the best one can obtain in view of the counterexamples of Villegas. Building on this result we then show that the singular set has dimension n-10, at least for a large class of nonlinearities that includes the most relevant cases. This is a joint work with Alessio Figalli.
  
  

• November 8, 2024: Alec Payne (SLMath), A Generalized Avoidance Principle for Mean Curvature Flow

  The avoidance principle says that two mean curvature flows of hypersurfaces remain disjoint if they are disjoint initially. In this talk, we will discuss several generalizations of the avoidance principle that allow for intersections between the two hypersurfaces. First, we show that the Hausdorff dimension of the intersection of two mean curvature flows is non-increasing over time. We then prove a dimension monotonicity result for self-intersections of immersed mean curvature flows. Next, we extend the intersection dimension monotonicity to a class of Brakke flows and level set flows, and we provide examples showing that this monotonicity fails for general Brakke flows. In the course of the proof, we find an equivalence between non-fattening and non-discrepancy for level set flows with finitely many singularities, and we explore the connection between the generalized avoidance principle and uniqueness of the flow through singularities. This is joint work with Tang-Kai Lee.
  
  

• November 15, 2024: Ao Sun (Lehigh University), Local dynamics and shape of mean curvature flow passing through a singularity

  A central question in geometric flow is to understand how the geometry and topology change after passing through singularities. I will explain how the local dynamics influence the shape of the flow near a singularity, and how the geometry and topology of the flow will change after passing through a singularity with generic dynamics. This talk is based on joint work with Zhihan Wang and Jinxin Xue.
  
  

• November 22, 2024 in room 417 (NOTICE THE ROOM CHANGE): Sigurd Angenent (University Wisconsin), The unstable manifold of Bohm solitons

  Bohm constructed Ricci flat asymptotically conical metrics on RkxSl, which are fixed points for Ricci flow. I will explain how, in joint work with Dan Knopf, we showed that any given Bohm metric gB is dynamically unstable, and that there is an infinite dimensional family of ancient solutions that converge to gB in backward time.
  
  

• December 6, 2024: Xuantao Chen (John Hopkins University), Regularity of the future event horizon in perturbations of Kerr

  In this talk, we study the regularity of the future event horizon in the perturbed spacetimes constructed by Klainerman-Szeftel and Giorgi-Klainerman-Szeftel in their proof of Kerr stability for small angular momentum. The proof we present can be extended to the full subextremal case, provided a corresponding stability result is established. This is joint work with Sergiu Klainerman.
  
  

• December 13, 2024: Allen Fang (Muenster University), TBA

  TBA
  
  

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Here are our past seminars in Spring 2024

• 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