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Schedule

Monday, July 13


  • 09:30 AM - 10:00 AM | Welcome Breakfast

  • 10:00 AM - 10:50 AM | Stan Palasek, IAS/Princeton University

    • Title: TBC
    Abstract: TBC


  • 11:00 AM - 11:50 AM | Bjoern Bringmann, Princeton University

    • Title: Construction of the 2D Yang-Mills-Higgs measure.
    Abstract: We discuss the construction of the 2D Yang-Mills-Higgs (YMH) measure via stochastic quantization. To do so, we show global well-posedness and uniform-in-time bounds for the associated Langevin dynamics, given by the 2D stochastic YMH equations. A key component of our approach is the further development of techniques in stochastic geometric analysis, combining ideas from geometric analysis and stochastic analysis. These methods yield a manifestly gauge-covariant local existence theory, refined estimates for covariant stochastic objects, and a decay mechanism driven by unstable Yang-Mills connections. Based on joint work with S. Cao, M. Hairer, and W. Zhao.


  • 12:00 PM - 02:00 PM | Lunch Break

  • 02:00 PM - 02:50 PM | Charles Collot, CY Cergy Paris Universite

    • Title: Determination of the long-time behaviour of solutions to the two-dimensional Keller-Segel system with critical mass
    Abstract: This presentation will describe the dynamics of solutions to the parabolic-elliptic Keller-Segel system in two dimensions. A first behaviour is that the solution is global and converges to a forward self-similar solution; this occurs if the mass of the solution is less than the explicit ground state mass threshold of 8 pi [Blanchet-Dolbeault-Perthame and related works]. A second behaviour is that the solution blows up in finite time, which happens if the mass is more than 8 pi. Explicit examples are given by the collapse of a single stationary state [Herrero-Velazquez, Raphael-Schweyer, Collot-Ghoul-Masmoudi-Nguyen] or the collision of several collapsing stationary states [Collot-Ghoul-Masmoudi-Nguyen].

    The presentation will focus on the case of the critical mass of 8 pi, for solutions with finite second moment. Such solutions are known to be global and to concentrate in infinite time [Carrillo-Masmoudi]. Moreover, based on physical predictions [Chavanis-Sire] an explicit example of such solution concentrating a stationary state at a scale log(t)^(-1/2) was constructed in the radial class [Ghoul-Masmoudi] and then in the non-radial class [Davila-del Pino-Dolbeault-Musso-Wei]. The main result that will be presented shows that this is the universal dynamics at critical mass, i.e. that all general solutions will converge to a renormalized stationary state which concentrates at scale log(t)^(-1/2) around the center of mass of the solution. The proof combines soliton resolution techniques, with the control of the evolution of the solution in various critical spaces, and with the perturbative analysis around an approximate solution involving multiple scales. This is joint work to appear with Federico Buseghin (CY Cergy Paris Universite)


  • 03:00 PM - 03:30 PM | Afternoon Coffee Break

  • 03:30 PM - 04:20 PM | Gong Chen, Georgia Tech

    • Title: Multi-channel distorted Fourier analysis and stability of one-dimensional multi-solitons
    Abstract: Distorted Fourier transforms diagonalize perturbed Schrodinger operators and play a central role in the analysis of asymptotic stability for a single soliton. In the multi-soliton problem, however, linearization around a train of moving coherent structures leads instead to an equation with several localized potentials moving at distinct velocities, commonly known as a charge-transfer model. Thus, to prove asymptotic stability of multi-solitons, one needs robust dispersive and localized decay estimates for such multi-potential linear flows. In this talk, I will focus on the one-dimensional setting, where the main linear difficulty is no longer the construction of a single distorted Fourier transform, but rather the need to use several moving distorted Fourier charts and glue them consistently across the regions between solitons. I will describe a multi-channel scattering construction for one-dimensional charge-transfer models. As an application, I will discuss the asymptotic stability problem for multi-solitons of the one-dimensional L^2-supercritical nonlinear Schrodinger equation. The linear charge-transfer theory provides dispersive and localized decay estimates for radiation in moving multi-potential backgrounds. Combined with modulation equations and a finite-dimensional center-stable condition that removes the unstable modes of the supercritical solitons, this yields asymptotic stability of multi-solitons.


  • 04:30 PM - 05:20 PM | Haitian Yue, ShanghaiTech

    • Title: Invariant Gibbs measures and propagation of randomness for nonlinear dispersive PDE
    Abstract: The study of propagation of randomness in the context of dispersive PDEs can be traced back to work by Lebowitz-Rose-Speer (1988, 1989) and Bourgain (1994, 1996) concerning the Gibbs measure for nonlinear Schrodinger equations. Since then there have been substantial developments of their ideas by many different researchers. In the last few years, this field has seen significant progress and many new ideas and methods have been introduced. The aim of this talk is to briefly review these recent developments (random averaging operators and random tensor theories) and describe some of the foundations upon which these recent developments have built upon, in particular Bourgains seminal work in the subject.


Tuesday, July 14


  • 09:30 AM - 10:00 AM | Welcome Breakfast

  • 10:00 AM - 10:50 AM | Jonathan Luk, Stanford University

    • Title: The formation of a weak null singularity in the interior of a generic rotating black hole
    Abstract: Given smooth data for the Einstein vacuum equations representing a dynamical event horizon settling down to that of Kerr in the subextremal, strictly rotating range with suitable upper and lower bounds, we prove that a weak null singularity forms, across which the spacetime metric is continuously extendible but not Lipschitz extendible. This is a joint work with Jan Sbierski.


  • 11:00 AM - 11:50 AM | Juhi Jang, University of Southern California

    • Title: Stable singularities in self-gravitating fluids
    Abstract: I will discuss recent progress on mathematical construction of self-similar solutions to the Euler-Poisson system describing gravitational collapse and nonlinear stability of the Larson-Penston collapse against radially symmetric perturbations. At the heart of the latter stability result is the ground state character of the Larson-Penston solution featuring important monotonicity properties. The talk is based on joint works with Yan Guo, Mahir Hadzic and Matthew Schrecker.


  • 12:00 PM - 02:00 PM | Lunch Break

  • 02:00 PM - 02:50 PM | Igor Rodnianski, Princeton University

    • Title: Critical collapse in 2+1 gravity
    Abstract: I will discuss joint work with S. Cicortas addressing existence of naked singularities in 2+1 dimensional Einstein equations with negative cosmological constant which turn out to describe a transition from a non-collapsing to a black hole forming regime.


  • 03:00 PM - 03:30 PM | Afternoon Coffee Break

Wednesday, July 15


  • 09:30 AM - 10:00 AM | Welcome Breakfast

  • 10:00 AM - 10:50 AM | Gunther Uhlmann, University of Washington

    • Title: Seeing Through Space-Time
    Abstract: We will consider the question on whether we can determine the structure of space time by making measurements near the worldline of an observer. We will consider both active and passive measurements. For the case of passive measurements one measures the fronts of light sources near the observer. For the case of active measurements we couple Einstein equations with matter or electromagnetic fields and formulate the question of determining the structure of space time as the problem of recovering the metric from observations of waves near the observer. This method applied to several other inverse problems in nonlinear wave propagation including nonlinear acoustics, nonlinear elasticity, fluid mechanics etc..


  • 11:00 AM - 11:50 AM | Katya Krupchyk, University of California Irvine

    • Title: Fractional Anisotropic Calder'on Problem
    Abstract: The anisotropic Calder'on problem asks whether a Riemannian metric, or more generally a compact Riemannian manifold with boundary, can be recovered from the Dirichlet-to-Neumann map for the Laplace-Beltrami operator, given on the boundary of the manifold, and it remains open in general for smooth metrics in dimensions three and higher. In this talk, we discuss a nonlocal analogue, the fractional anisotropic Calder'on problem, and present uniqueness results in two settings. First, on a smooth closed Riemannian manifold, we show that the source-to-solution map for the fractional Laplace-Beltrami operator, known on an arbitrary nonempty open subset, determines the manifold up to isometry. Second, in Euclidean space, we demonstrate that the partial exterior Dirichlet-to-Neumann map for the fractional Laplace-Beltrami operator, known on an arbitrary nonempty open subset of the exterior of the domain, determines the smooth Riemannian metric up to a diffeomorphism fixing the exterior, for metrics that agree with the Euclidean metric outside a compact set. The talk is based on joint works with Ali Feizmohammadi, Tuhin Ghosh, Angkana Ruland, Johannes Sjostrand, and Gunther Uhlmann.


  • 12:00 PM - 02:00 PM | Lunch Break

  • 02:00 PM - 02:50 PM | Jingni Xiao, Drexel University

    • Title: Non-Scattering Phenomena for Inhomogeneous Media
    Abstract: In this talk I will describe several PDE techniques for studying non-scattering, including scattering by corner and edge singularities via complex geometrical optics (CGO) solutions, regularity of non-scattering interfaces through free boundary methods, and finiteness results for non-scattering wave numbers. These results show that the geometry and regularity of an inhomogeneous medium strongly constrain the possibility of invisibility, bringing together ideas from elliptic PDE, spectral theory, and inverse scattering.


  • 03:00 PM - 03:30 PM | Afternoon Coffee Break

  • 03:30 PM - 04:20 PM | Alexandru Ionescu, Princeton University

    • Title: On the non-uniqueness of solutions of the swirl-free axi-symmetric Navier-Stokes equations
    Abstract: I will discuss some recent computer-assisted work on the question of non-uniqueness of solutions of the incompressible Navier-Stokes in 3D. The incompressible Navier-Stokes equations are "regular" equations in the class of axi-symmetric swirl-free solutions, for which large data global regularity of smooth solutions is known. The point of the lecture is to provide evidence that non-uniqueness of solutions still holds in critical and slightly super-critical spaces. This is joint work with Hao Jia and Stan Palasek.


  • 04:30 PM - 05:20 PM | Sanchit Chaturvedi, New York University

    • Title: Shock formation in 2D Euler in presence of a wall with no penetration boundary conditions
    Abstract: In recent years, shock formation for Euler equations in higher dimensions has attracted a lot of attention and the field has entered a fairly mature stage. The field builds on geometric methods that have found a lot of success in general relativity but due to a severe dependence of the acoustical metric on the gas, novel ideas were needed to simultaneously treat the emergence of the severe singularity and the loss of derivatives that the geometric approach causes. In this talk I will discuss the problem of shock formation in 2D Euler gas with a wall with no penetration boundary conditions such that the wall plays a nontrivial role in the analysis. Due to the characteristics bouncing off of the wall, the usual ideas employed for shock formation fail at a fundamental level. I will talk about the geometric set up employed in the free space case, then explain why that set up must fail in our case and then finally discuss a geometric set up that does work. This is based on an ongoing joint work with John Anderson.


Thursday, July 16


  • 09:30 AM - 10:00 AM | Welcome Breakfast

  • 10:00 AM - 10:50 AM | Gigliola Staffilani, Massachusetts Institute of Technology

    • Title: Some mathematically rigorous results in wave turbulence theory
    Abstract: In this talk we give an overview of some new mathematical advances in wave turbulence theory. We start with the original perspective of Bourgain on the study of the energy spectrum for a periodic nonlinear Schrodinger (NLS) equation via growth of Sobolev norms. Here we present an almost dichotomy depending on the specific periodic conditions of the NLS problem. Then we move to the wave kinetic equations as effective equations for the energy spectrum itself and we give an example of a rigorous proof for energy transfer. We end with an implosion result for a periodic defocusing NLS equation.


  • 11:00 AM - 11:50 AM | Natasa Pavlovic, University of Texas Austin

    • Title: A tale of two Landau equations
    Abstract: We will present results on two different versions of Landau equation. In the first part of the talk we will investigate a fuzzy variant of the inhomogeneous Landau equation and establish global-in-time existence and uniqueness of smooth solutions for a range of soft potentials. The spatial delocalization introduced in the collision operator not only prevents singularity formation and enhances regularity, but also reveals additional structural properties of the model. This part of the talk is based on the joint work with Maria Pia Gualdani, Nestor Guillen, Maja Taskovic and Nicola Zamponi. The second part of the talk will focus on deriving the quantum Landau operator as the weak-coupling limit of the quantum Boltzmann operator (also known as the Uehling-Uhlenbeck operator). We consider both Fermi-Dirac and Bose-Einstein statistics. Our approach is inspired by the work by Benedetto and Pulvirenti, where the classical Landau operator was derived from the quantum Boltzmann operator. To capture the ternary term in the quantum Landau operator, we introduce a new two-parameter scaling that preserves the quantum effects in the limit. This part of the talk is based on the joint work with Maria Pia Gualdani, Justin Toyota and Dominic Wynter.


  • 12:00 PM - 02:00 PM | Lunch Break

  • 02:00 PM - 02:50 PM | Sylvia Serfaty, New York University

    • Title: Two methods for deriving singular mean-field limits
    Abstract: We are interested in the question of mean-field limits, or deriving effective evolution equations of PDE type for a system of N points in singular interaction, for instance of Coulomb or Riesz nature, evolving by first order dynamics. We will discuss two methods: the modulated energy method, that works well for gradient flows or conservative flows of Coulomb/Riesz type energies, and a new method based on a multiscale mollification metric, which works well for up to Coulomb interaction singularity, without much structure assumed.


  • 03:00 PM - 03:30 PM | Afternoon Coffee Break

  • 03:30 PM - 04:20 PM | Sohrab Shahshahani, University of Massachusetts Amherst

    • Title: Asymptotic Stability of the Degree-One Vortex in the Abelian Yang-Mills-Higgs Model
    Abstract: The abelian Yang-Mills-Higgs equations in two space dimensions admit topological vortex solutions. I will discuss the asymptotic stability of the degree-one vortex at self-dual coupling, for small equivariant perturbations. An important feature of the problem is the presence of an internal mode in the spectral gap of the linearized operator. The interaction between this discrete mode and the continuous spectrum leads to a damping mechanism, quantified by a nonlinear Fermi Golden rule. The main difficulties in the proof of stability are 1) The slow decay of the internal mode. 2) Proving decay for the projection of the perturbation onto the continuous spectrum, which satisfies a Klein-Gordon equation with a matrix potential. The latter is achieved by combining the spacetime resonance method for the flat Klein-Gordon equation with local energy decay for the linearized operator. This is joint work with Jose Palacios, Fabio Pusateri, Jonas Luhrmann, and Wilhelm Schlag.