Talks (titles and abstracts)
Barry Barish: Gravitational Wave Observations and Implications
Lydia Bieri: Insights into the Gravitational Wave Memory Effect
A major breakthrough of General Relativity (GR) happened in 2015 with LIGO's first detection of gravitational waves. In Mathematical GR the Einstein equations describe the laws of the universe. This system of hyperbolic nonlinear pde hasserved as a playground for all kinds of new problems and methods in pde analysis and geometry.
Gravitational radiation carries information about the sources such as mergers of binary black holes, binary neutron stars and core-collapse supernovae. In these processes mass and momenta are radiated away in form of gravitational waves. GR predicts that these waves leave a footprint in the spacetime, that is they change the spacetime permanently, which results in a permanent displacement of test masses. This effect is called the memory. In this talk, I will explore the gravitational wave memory. We will see that there are two types of memory, one going back to Ya. B. Zel'dovich and A. G. Polnarev and one to D. Christodoulou. I will also discuss recent work with D. Garfinkle and N. Yunes on cosmological memory.
Claudio Bunster: Gravitational domain walls and the dynamics of G
From the point of view of elementary particle physics the gravitational constant G is extraordinarily small. This has led to ask whether it could have decayed to its present value from an initial one commensurate with microscopical units. A mechanism that leads to such a decay, based on the possibility that G may take take different values within regions of the universe separated by a domain wall, is explored.
Gui-Qiang Chen: On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type in Mechanics and Geometry
As is well-known, two of the basic types of linear PDEs are elliptic type and hyperbolic type, following the classification of linear PDEs first proposed by Jacques Hadamard in the 1920s; and linear theories of PDEs of these two types have been considerably established, respectively. On the other hand, many nonlinear PDEs arising in mechanics, geometry, and other areas naturally are of mixed elliptic-hyperbolic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear PDEs of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear PDEs of mixed elliptic-hyperbolic type with these longstanding problems and will then discuss some of the most recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing and identifying mathematical approaches, ideas, and techniques for dealing with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.
Mihalis Dafermos: Christodoulou as Cosmic Censor
The cosmic censorship conjectures constitute the most central unsolved problems in classical general relativity. I will describe how these conjectures were originally posed by Penrose, and how their formulations were transformed by the seminal work of Demetrios Christodoulou, who in the 1980s and 90s uncovered surprising examples of naked singularities yet succeeded in proving definitive versions of cosmic censorship for the collapse of a spherically symmetric self-gravitating scalar field. Christodoulou’s work remains today the source of our best intuition regarding the validity of these conjectures. I will end with a brief discussion of some more recent extensions and open prospects for the future.
Thibault Damour: Gravitational Waves from Coalescing Black Holes
The recent discovery of several gravitational wave events by the two Laser Interferometer Gravitational-Wave Observatory (LIGO) interferometers has brought the first direct evidence for the existence of black holes, and has also been the first observation of gravitational waves in the wave-zone. The talk will review the theoretical developments on the motion and gravitational radiation of binary black holes that have been crucial in interpreting the LIGO events as being emitted by the coalescence of two black holes. In particular, we shall present the Effective One-Body (EOB) formalism which led to the first prediction for the gravitational-wave signal emitted by coalescing black holes. After a suitable Numerical-Relativity completion, the (analytical) EOB formalism has allowed one to compute the bank of 200 000 accurate templates that has been used to search the first coalescence signals, and to measure the masses and spins of the coalescing black holes.
Ruth Durrer: General Relativity and Cosmology
Cosmology cannot really be formulated without General Relativity (GR). Only within the GR framework can we consider a consistent gravitating spacetime. After a historical perspective I will concentrate on a few ’pearls’ of present cosmological research like the cosmic microwave background (CMB), weak lensing and weak lensing of the CMB. I shall explain how planned high precision CMB polarisation experiments will be able to measure effects of frame dragging on cosmological scales.
Domenico Giulini: Aspects of 3-manifold theory motivated by General Relativity
Einstein's vacuum equations may be cast into the form of a constrained Hamiltonian system in which the object undergoing dynamical evolution is a Riemannian geometry on a 3-manifold, the topology of which may be freely specified. Hamiltonian reduction shows that under certain conditions the 3d mapping-class group is isomorphic to the fundamental group of the 3d Riemannian moduli space, the determination of which is helped by "physical intuition" in an interesting fashion.
Gerhard Huisken: Foliations in Asymptotically flat 3-manifolds
Asymptotically flat 3-manifolds arise as space-like slices in Lorentzian models of isolated gravitating systems in General Relativity. Radial foliations by 2-dimensional surfaces in such 3-manifolds satisfying some geometric variational principle provide structures that allow a description of physical concepts such as mass, quasi-local mass, center of mass, momentum in a geometric way independent of coordinate systems. The lecture describes several such foliations and their properties.
Carlos Kenig: The energy channels method
We will explain the method of "channels of energy" and some of its applications to study singularity formation for some nonlinear wave equations.
Sergiu Klainerman: Legacy of the proof of the Nonlinear Stability of Minkowski space
I will be talking about the impact of the proof on several directions. Of research in General Relativity and Nonlinear Wave Equations.
Joachim Krieger: Recent developments in the theory of Wave Maps
I will explain a number of recent results about regularity and singularity formation of critical Wave Maps and the role that by now classical work by Christodoulou-Shatah -Tahvildar-Zadeh played in their genesis.
Enno Lenzmann: Heisenberg, Haldane and Half-Wave Maps
The half-wave maps equation is a novel geometric evolution equation, which displays many intriguing mathematical properties with links to minimal surfaces, conformal symmetry, and completely integrable systems. In this first part of my talk, I will highlight the physical motivation for this evolution equation from exactly solvable quantum systems (Haldane-Shastry models) and completely intergable Calogero-Moser spin systems. In the second part of my talk, I will discuss recent results about the explicit classification of traveling solitary waves together with their complete spectral analysis. This is joint work with Patrick Gérard and Armin Schikorra.
Hans Lindblad: Global stability of Minkowski space for the Einstein--Vlasov system in the harmonic gauge
Minkowski space is shown to be globally stable as a solution to the massive Einstein--Vlasov system. The proof is based on a harmonic gauge, in which the Einstein equations reduce to a system of quasilinear wave equations satisfying the weak null condition. Central to the proof is a collection of vector fields, used to control the Vlasov part of the solutions, which are adapted to the solution and reduce to the generators of the symmetries of Minkowski space when restricted to acting on spacetime functions. This is joint work with Martin Taylor.
Andrew Majda: Low-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification in Turbulent Systems
Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering including climate, material, and neural science. The existence of a strange attractor in the turbulent systems containing a large number of positive Lyapunov exponents results in a rapid growth of small uncertainties from imperfect modeling equations or perturbations in initial values, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification (UQ) in turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in their nonlinear responses to changes in external forcing parameters or uncertain initial data. In the development of a proper UQ scheme for systems of high or infinite dimensionality with instabilities, significant model errors compared with the true natural signal are always unavoidable due to both the imperfect understanding of the underlying physical processes and the limited computational resources available through direct Monte-Carlo integration. One central issue in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity).
Here we discuss a general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems. There are three stages in the modeling strategy, imperfect model selection; calibration of the imperfect model in a training phase using only data in the more complex perfect model statistics; and prediction of the responses with UQ to a wide class of forcing and perturbation scenarios. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. Several mathematical ideas will be introduced to improve the prediction skill of the imperfect reduced-order models. Most importantly, empirical information theory and statistical linear response theory are applied in the training phase for calibrating model errors to achieve optimal imperfect model parameters; and total statistical energy dynamics are introduced to improve the model sensitivity in the prediction phase especially when strong external perturbations are exerted. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed. The uncertainty quantification and statistical response for these complex models are accurately captured by the reduced-order models with only 2x10^2 modes in a highly turbulent system with 1x10^5 degrees of freedom. Less than 0.15% of the total spectral modes are needed in the reduced-order models.
Wilhelm Schlag: On longterm dynamics of dispersive evolution equations
Richard Schoen: The high dimensional positive energy theorem
We will discuss the problem of proving the positive energy theorem in cases not directly covered by the Dirac operator approach or the minimal hypersurface approach. These are the cases when the dimension is greater than 8 and the manifold is not spin. The proof is accomplished by extending the minimal hypersurface approach in the presence of singularities and controlling the singular sets which arise. The work is in a recent paper which is joint with S. T. Yau.
Jalal Shatah: Weak Turbulence
David Spergel: Measuring the geometry and topology of the universe with CMB observations
Over the past twenty years, astronomers have made precision measurements of the microwave background. I will discuss how these observations constrain the universe’s shape and show that the universe is nearly flat and very large.
A. Shadi Tahvildar-Zadeh: General Relativity at the Atomic Scale
"To what extent can general relativity account for atomic structure of matter and for quantum effects?" This was a question asked by Einstein and Rosen in their famous paper of 1935 entitled "The Particle Problem in the General Theory of Relativity" (nowadays widely referred to as "the ER paper"). More recently, prominent physicists such as Freeman Dyson have cast doubt on the possibility, or even the wisdom of, combining general relativity with quantum mechanics. In this talk I will describe how our knowledge of general relativity can in fact help us gain new insight into the microscopic world of elementary particles and their quantum laws of motion. I will revisit Herman Weyl's so-called "singularity theory" of matter, and explain that the main obstacle on the path to a possible symbiosis of GR and QM is a well-known classical problem, namely the infinities inherent in Maxwell-Lorentz electrodynamics of point charges. I will discuss a possible remedy to this problem, and report on the recent progress my colleague Michael Kiessling and I have made in this research project, which also involves some of our students and research associates.
Robert M. Wald: You Can't Over-Charge or Over-Spin a Black Hole
The Kerr-Newman solutions are the only stationary black hole solutions of the Einstein-Maxwell equations in 4-dimensions. However, these solutions describe black holes only when the inequality \(M^2 \geq (J/M)^2 + Q^2\) is satisfied, where \(M\), \(J\), and \(Q\) are the mass, angular momentum, and charge of the black hole. Therefore, if an extremal or nearly extremal black hole can be made to absorb matter with sufficiently large angular momentum or charge as compared with its energy, one would obtain an apparent contradiction with cosmic censorship. Hubeny and others have made proposals as to how this might be done, but a proper analysis of this proposal requires a calculation of all second order effects on energy, including, in particular, effects arising from self-force. We show in this work that when all of the second order effects are taken into account, no over-charging or over-spinning of a black hole can occur, provided only that the non-electromagnetic matter satisfies the null energy condition.
Gilbert Weinstein: Harmonic maps with prescribed singularities and applications to general relativity
Several decades ago, harmonic maps with prescribed singularities proved to be useful in investigations of the Einstein vacuum and Einstein Maxwell equations under the hypothesis of stationarity and axial symmetry. More recently, new applications have been found including lower bounds on the total mass, as well as lower bounds on the area, in terms of angular momentum and charge, for axially symmetric asymptotically flat data. In our most recent work, joint with M. Khuri and S. Yamada, we use similar ideas to study stationary bi-axially symmetric black hole solutions with non-standard horizon topology for the 5-dimensional Einstein vacuum equations. In this talk, I will survey these applications of harmonic maps to general relativity with somewhat more emphasis on the latter work.
Sijue Wu: On two dimensional water waves with angled crests
I will discuss recent work on the wellposedness of the Cauchy problem of the two dimensional water wave equation in a regime that includes water waves with non-\(C^1\) interfaces.
Zhouping Xin: Transonic Shocks in Curved Nozzles
In this talk, I will discuss some steady compressible flows in nozzles with variable cross sections. We will focus on flows with shocks with physical boundary conditions. In particular, I will present some results on the Courant-Friedrich's transonic shock problem in some classes of general 2-dimensional nozzles. This will be a nonlinear free boundary value problem with nonlinear boundary conditions for mixed type equations. Existence of single or multiple transonic shocks will be discussed in terms of the geometry of the nozzle and the given exit pressure. Some ideas of analysis will be presented.
Shing-Tung Yau: General relativity and important physical quantities