Caltech Analysis Seminar

Spring 2021

April 15

Toan Nguyen (Penn State University)

Title: On the stability of classical boundary layers

Abstract: The talk is to give an overview on the (in)stability of classical boundary layers in fluid dynamics that were introduced by Prandtl in 1904 to approximate viscous flows near a solid boundary in the small viscosity limit. In particular, near boundary layers with an inflection point, the classical Ansatz is incorrect due to an emergence of thinner viscous sublayers, which could remarkably reach to an order one in its amplitude. Near generic stable boundary layers, these sublayers may still become unstable due to the classical viscous destabilization phenomena. The talk is based on a series of recent joint works with E. Grenier (ENS Lyon).

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Winter 2021

January 7

Robin Neumayer (Northwestern University)

Title: Quantitative stability for minimizing Yamabe metrics

Abstract: The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric. We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show, in a quantitative sense, that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

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January 14

Pierre Germain (NYU)

Title: Stability of kinks in one-dimensional Klein-Gordon equations

Abstract: Kinks are topological solitons, which appear in (nonlinear) one-dimensional Klein-Gordon equations, the Phi-4 and Sine-Gordon equations being the most well-known examples. I will present new results which give asymptotic stability for kinks, with an optimal decay rate, in some cases. The proof relies on the distorted Fourier transform associated to the linearized equation around the kink; this method should be of interest for more general soliton stability problems. This is joint work with Fabio Pusateri.

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January 21

Zeev Rudnick (Tel Aviv University)

Title: The Robin eigenvalue problem: statistics and arithmetic

Abstract: Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. The focus of the lecture will be the fluctuations of the gaps between the Robin and Neumann eigenvalues for planar domains. I will present some numerical experimentation which reveals connections with number theory and quantum ergodicity, some new results inspired by these numerical explorations, and several open problems. Joint work with Igor Wigman and Nadav Yesha.
A couple of recent preprints are:
Zeev Rudnick, Igor Wigman, Nadav Yesha. Differences between Robin and Neumann eigenvalues https://arxiv.org/abs/2008.07400
Zeev Rudnick and Igor Wigman On the Robin spectrum for the hemisphere. https://arxiv.org/abs/2008.12964

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January 28

Joseph Keir (Oxford University)

Title: Nonlinear wave equations, the weak null condition, and radiation fields.

Abstract: Nonlinear wave equations are ubiquitous in physics, and in three spatial dimensions they can exhibit a wide range of interesting behaviour even in the small data regime, ranging from dispersion and scattering on the one hand, through to finite-time blowup on the other. The type of behaviour exhibited depends on the kinds of nonlinearities present in the equations. In this talk I will explore the boundary between "good" nonlinearities (leading to dispersion similar to the linear waves) and "bad" nonlinearities (leading to finite-time blowup). In particular, I will give an overview of a proof of global existence (for small initial data) for a wide class of nonlinear wave equations, including some which almost fail to exist globally, but in which the singularity in some sense takes an infinite time to form. I will also show how to construct other examples of nonlinear wave equations whose solutions exhibit very unusual asymptotic behaviour, while still admitting global small data solutions.

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February 4

Hugo Duminil-Copin (IHES)

Title: Emergent symmetries in statistical physics systems

Abstract: A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millenium, the mathematical understanding of this fact has progressed tremendously in two dimensions with the introduction of the Schramm-Loewner Evolution and the proofs of conformal invariance of the Ising model and dimers. Nevertheless, the understanding is still restricted to very specific models. In this talk, we will gently introduce the notion of conformal invariance of lattice systems by taking the example of percolation models. We will also explain some recent and partial progress in the direction of proving conformal invariance for a large class of such models.

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February 11

Mikhail Karpukhin (Caltech)

Title: Eigenvalues of the Laplacian and min-max for energy functional.

Abstract: The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of optimal upper bounds for its eigenvalues is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. It turns out that the optimal isoperimetric inequalities for Laplacian eigenvalues are closely related to minimal surfaces and harmonic maps. In the present talk we survey recent developments in the field. In particular, we will discuss a min-max construction for the energy functional and its applications to eigenvalue inequalities, including the regularity theorem for optimal metrics. The talk is based on the joint work with D. Stern.

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February 18

Salvatore Stuvard (UT Austin)

Title: Mean curvature flow with prescribed boundary: a dynamical approach to Plateau's problem

Abstract: The Brakke flow is a measure-theoretic generalization of the mean curvature flow which describes the evolution by mean curvature of surfaces with singularities. In the first part of the talk, I am going to discuss global existence and large time asymptotics of solutions to the Brakke flow with fixed boundary when the initial datum is given by any arbitrary rectifiable closed subset of a convex domain which disconnects the domain into finitely many "grains". Such flow represents the motion of material interfaces constrained at the boundary of the domain, and evolving towards a configuration of mechanical equilibrium according to the gradient of their potential energy due to surface tension. In the second part, I will focus on the case when the initial datum is already in equilibrium (a generalized minimal surface): I will prove that, in presence of certain singularity types in the initial datum, there always exists a non-constant solution to the Brakke flow. This suggests that the class of dynamically stable minimal surfaces, that is minimal surfaces which do not move by Brakke flow, may be worthy of further study within the investigation on the regularity properties of minimal surfaces. Based on joint works with Yoshihiro Tonegawa (Tokyo Institute of Technology).

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February 25

Cécicle Huneau (École Polytechnique)

Title: High frequency limit for Einstein equations with a U(1) symmetry.

Abstract: Due to the nonlinear character of Einstein equations, a sequence of metrics, solutions in vacuum, which oscillate with higher and higher frequency may converge to a solution to Einstein equations coupled to some effective energy momentum tensor. This effect is called backreaction, and has been studied by physicists (Isaacson, Burnett, Green and Wald). It has been conjectured by Burnett, under some definition of the high frequency limit, that the only effective energy momentum tensor that could appear corresponds to a massless Vlasov field, and that reciprocally all solutions to Einstein equations coupled to a massless Vlasov field can be approached by a sequence of solutions to Einstein vacuum equations (with higher and higher frequencies oscillations). I will present a work in collaboration with Jonathan Luk around Burnett conjecture and its reverse.

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March 4

John Anderson (Princeton University)

Title: Stability results for anisotropic systems of wave equations

Abstract: In this talk, I will describe a global stability result for a nonlinear anisotropic system of wave equations. This is motivated by studying phenomena involving characteristics with multiple sheets. For the proof, I will describe a strategy for controlling the solution based on bilinear energy estimates. Through a duality argument, this will allow us to prove decay in physical space using decay estimates for the homogeneous wave equation as a black box. The final proof will also require us to exploit a certain null condition that is present when the anisotropic system of wave equations satisfies a structural property involving the light cones of the equations.

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March 11

Monica Visan (UCLA)

Title: Orbital stability of KdV multisolitons in $H^{-1}$

Abstract: We introduce a variational characterization of multisoliton solutions to the Korteweg-de Vries equation that is meaningful in $H^{-1}$, which is the space of optimal well-posedness for this equation. As a consequence, we obtain orbital stability of multisoliton solutions in $H^{-1}$. This is joint work with Rowan Killip.

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Fall 2020

October 22

Maxime van de Moortel (Princeton University)

Title: Nonlinear interaction of three impulsive gravitational waves for the Einstein equations

Abstract: An impulsive gravitational wave is a weak solution of the Einstein vacuum equations whose metric admits a curvature delta singularity supported on a null hypersurface; the spacetime is then an idealization of a gravitational wave emanating from a strongly gravitating source. In the presence of multiple sources, their impulsive waves eventually interact and it is interesting to study the spacetime up to and after the interaction. For such singular solutions, the classical well-posedness results (such as the bounded L^2 curvature theorem) are not applicable and it is not even clear a priori whether the initial regularity propagates or a worse singularity occurs from the nonlinear interaction. I will present a local existence result for U(1)-polarized Cauchy data featuring three impulsive gravitational waves of small amplitude propagating towards each other. The proof is achieved with the help of localization techniques inspired from Christodoulou's short pulse method and new tools in Harmonic Analysis, notably anisotropic estimates that are tailored to the problem. This is joint work with Jonathan Luk.

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October 29

Dejan Gajic (University of Cambridge)

Title: Mode coupling and late-time asymptotics of linear waves on rotating black holes

Abstract: I will discuss a recently discovered connection between the presence of late-time tails in the dynamical behavior of black hole solutions to Einstein's equations of general relativity and the existence of conservation laws "at infinity". Understanding late-time tails is important for determining the nature of singularities inside dynamical black holes. I will focus in particular on new work, obtained in collaboration with Y. Angelopoulos and S. Aretakis, that addresses the effect of the rotation of black holes on late-time tails.

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November 5

Nicolas Rougerie (Grenoble)

Title: Non-linear Gibbs measures as mean-field limits

Abstract: To certain non-linear Schrödinger (NLS) equations, one can associate an invariant Gibbs measure based on the conserved energy. This is the basic ingredient of the Euclidean approach to constructive quantum field theory, as well as the large-time asymptote for the stochastic non-linear heat equation. I shall discuss a certain mean-field limit relating these Gibbs measures and the positive-temperature equilibria of the underlying many-body model (the Bose gas). A difficulty is that the Gibbs measure lives on low regularity distributional spaces, so that the non-linearity has to be understood in a renormalized sense. I shall put emphasis on the control of the renormalization procedure at the level of the quantum many-body model. joint work with Mathieu Lewin and Phan Thành Nam

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November 12

Shukun Wu (UIUC)

Title: On the Bochner-Riesz operators and the maximal Bochner-Riesz operator.

Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we use are polynomial partitioning, and the Bourgain Demeter l^2 decoupling theorem.

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November 19

Henrik Matthiesen (University of Chicago)

Title: New minimal surfaces from shape optimization.

Abstract: I will discuss the connection between sharp eigenvalue bounds and minimal surfaces in two cases: The first eigenvalue of the Laplacian on a closed surface among unit area metrics, and the first Steklov eigenvalue on a compact surface with non empty boundary among metrics with unit length boundary. In both cases maximizing metrics - if they exist - are induced by certain minimal immersions. More precisely, minimal immersions into round spheres for the closed case and free boundary minimal immersions into Euclidean balls in the bordered case. I will discuss the solution of the existence problem for maximizers in both these cases, which provides many new examples of minimal surfaces of the aforementioned types. This is based on joint work with Anna Siffert in the closed case and Romain Petrides in the bordered case.

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December 3

Juhi Jang (USC)

Title: Dynamics of Newtonian stars

Abstract: A classical model to describe the dynamics of Newtonian stars is the gravitational Euler-Poisson system. The Euler-Poisson system admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In this talk, I will discuss some recent progress on those star solutions with focus on expansion and collapse.

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December 10

Aron Wennman (Tel Aviv University)

Title: The hole event for Gaussian complex zeros and the emergence of quadrature domains

Abstract: The Gaussian Entire Function (GEF) is a distinguished random Taylor series with independent complex Gaussian coefficients, whose zero set is invariant with respect to isometries of the plane. The topic of this talk is the zero distribution of the GEF, conditioned on the event that no zero lies in a given (large) region. For circular holes Ghosh and Nishry observed that as the radius of the hole tends to infinity, the density of zeros vanishes not only on the given hole, but also on an annulus beyond the (rescaled) hole — a 'forbidden region' emerges. We are concerned with the shape of the forbidden region for general simply connected holes. I plan to discuss how one can study this problem through a type of constrained obstacle problem, and a curious emergence of quadrature domains. Based on joint work with Alon Nishry.

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