May 4, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Zoe Wyatt
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King's College London

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Title:
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Global stability of Kaluza-Klein spacetimes

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Abstract:
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Spacetimes formed from the cartesian product of Minkowski space and a flat torus play an important role as toy models for theories of supergravity and string theory. In this talk I will discuss an upcoming work, joint with Huneau and Stingo, showing the nonlinear stability of such a Kaluza-Klein spacetime. I will also explain how our result is connected to some earlier work of Witten.

April 27, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Susanna Haziot
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Brown University

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Title:
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The desingularization of small moving corners for the Muskat equation

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Abstract:
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The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. In this talk, we investigate the small data critical regularity theory for this equation, and in particular, the desingularization of interfaces with small moving corners. This is a joint work with Eduardo Garcia-Juarez (Universidad de Sevilla), Javier Gomez-Serrano (Brown University) and Benoit Pausader (Brown University).

April 20, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Hans Lindblad
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John Hopkins University

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Title:
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Modified Scattering for nonlinear wave equations

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Abstract:
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We construct solutions with prescribed radiation fields for wave equations with polynomially decaying sources close to the lightcone. In this setting, which is motivated by semi-linear wave equations satisfying the weak null condition, solutions to the forward problem have a logarithmic leading order term on the lightcone and non-trivial homogeneous asymptotics in the interior of the lightcone. The backward scattering solutions we construct from knowledge of the source and the radiation field at null infinity alone are given to second order by explicit asymptotic solutions which satisfy novel matching conditions close to the light cone. We also relate the asymptotics of the radiation field towards space-like infinity to explicit homogeneous solutions in the exterior of the light cone for slowly polynomially decaying data corresponding to mass, charge and angular momentum in the applications. The somewhat surprising discovery is that these data can cause the same logarithmic radiation field as the source term. This requires a delicate analysis of the forward homogeneous solution close to the light cone using the invertibility of the Funk transform.
This is joint work with Volker Schlue.

April 13, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Christoph Kehle
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ETH Zurich

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Title:
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Turbulence for quasilinear waves on Schwarzschild-AdS black holes

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Abstract:
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I will present upcoming joint work with Georgios Moschidis motivated by the question of black hole stability in the presence of a negative cosmological constant in the Einstein equations. We prove a ``weak turbulent instability’’ for quasilinear wave equations on generic Schwarzschild-AdS black holes. The instability is governed by a stably trapped 3-mode interaction transferring energy from low-to-high-frequency modes.

March 23, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Dejan Gajic
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Leipzig University

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Title:
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Stability and instability of extremal black holes

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Abstract:
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When Kerr black holes rotate at their maximally allowed angular velocity, they are said to be extremal. Extremal black holes are critical solutions to Einstein’s equations of general relativity. They exhibit interesting phenomena that are not present in more slowly rotating black holes. I will introduce recent work on the existence of strong asymptotic instabilities of a non-axisymmetric nature for scalar waves propagating on extremal Kerr black hole backgrounds and I will discuss the connection with previously known axisymmetric instabilities as well as with late-time power law tails in gravitational radiation.

March 2, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Gabriele Benomio
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Princeton University

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Title:
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A new gauge for gravitational perturbations of Kerr spacetimes

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Abstract:
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I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range. Central to the framework is a new formulation of nonlinear gravitational perturbations of Kerr in a geometric gauge tailored to the outgoing principal null geodesics of Kerr. The main features of the framework will be illustrated in the context of the linearised theory, which serves as a fundamental building block in nonlinear applications.

February 23, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Robert Strain
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University of Pennsylvania

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Title:
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Well-Posedness of the 3D Peskin Problem

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Abstract:
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This work introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. This work obtains the equations that model this free boundary problem and shows that these equations admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear 3D Peskin problem, and we prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time. This is a joint work with Eduardo García-Juárez, Po-Chun Kuo, and Yoichiro Mori.

February 16, 2023, Thursday, 2:00-3:00pm, Hill 705

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Speaker: Ryan Unger (double-header, part I)
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Princeton University & University of Cambridge

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Title:
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Retiring the third law of black hole thermodynamics

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Abstract:
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In this talk I will present a rigorous construction of examples of black hole formation which are exactly isometric to extremal Reissner–Nordström after finite time. In particular, our result can be viewed as a definitive disproof of the “third law of black hole thermodynamics.” This is joint work with Christoph Kehle.

February 16, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Yu Deng (double-header, part II)
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University of Southern California

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Title:
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Kinetic wave turbulence theory: full range of scaling laws

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Abstract:
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We present recent work with Zaher Hani (Michigan) that establishes the wave kinetic approximation for nonlinear Schrödinger (NLS) equation, for the full range of scaling laws that quantify the large box and weak nonlinearity limits. This completes the program, initiated in our earlier works, of providing a rigorous mathematical foundation for the wave turbulence theory. The proof involves refined analysis of very high order Feynman diagrams, including a new robust combinatorial algorithm and delicate cancellations between highly complicated diagrams that are identified as the result of this new algorithm.

February 9, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Wilhelm Schlag
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Yale University

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Title:
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On co-dimension one stability of the soliton for the 1D focusing cubic
Klein-Gordon equation

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Abstract:
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Solitons are particle-like solutions to dispersive evolution equations
whose shapes persist as time evolves. In some situations, these solitons
appear due to the balance between nonlinear effects and dispersion, in
other situations their existence is related to topological properties of
the model. Broadly speaking, they form the building blocks for the
long-time dynamics of dispersive equations.
In this talk I will present forthcoming joint work with J. Luehrmann (TAMU) on
long-time decay estimates for perturbations of the soliton for the 1D
focusing cubic Klein-Gordon equation (up to exponential time scales),
and I will discuss our previous work on the asymptotic stability of the
sine-Gordon kink under odd perturbations. While these two problems are
quite similar at first sight, we will see that they differ by a subtle
cancellation property, which has significant consequences for the
long-time dynamics of the perturbations of the respective solitons.

February 2, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Bjoern Bringmann
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Princeton University & IAS

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Title:
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Invariant Gibbs measures for the three-dimensional cubic nonlinear wave equation.

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Abstract:
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In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ43-model. This resultis the hyperbolic counterpart to seminal works on the parabolic Φ43-model by Hairer ’14 and Hairer-Matetski ’18. In the first half of this talk, we illustrate Gibbs measures in the context of Hamiltonian ODEs, which serve as toy-models. We also connect our theorem with classical and recentdevelopments in constructive QFT, dispersive PDEs, and stochastic PDEs. In the second half of this talk, we give a non-technical overview of the proof. As part of this overview, we first introduce a caloric representation of the Gibbs measure, which leads to an interplay of both parabolic and hyperbolic theories. Then, we discuss our para-controlled Ansatz and a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue.

January 26, 2023, Thursday, 3:30-4:30pm, Hill 705

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Speaker: Alexey Cheskidov
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UIC & IAS

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Title:
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Turbulent solutions of fluid equations

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Abstract:
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In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit pathological or wild behavior. These include non-uniqueness, singularity formation, the loss of energy balance, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness (uniqueness, regularity) results in bigger spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations.

December 8, 2022, Thursday, 2:00-3:00pm, Hill 525

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Speaker: Ruixiang Zhang
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UC Berkeley

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Title:
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Local smoothing for the wave equation in $2 + 1$ dimensions

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Abstract:
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Sogge's local smoothing conjecture for the wave equation predicts that the local $L^p$ space-time estimate gains a fractional derivative of order almost $1/p$ compared to the fixed time $L^p$ estimates, when $p>2n/(n-1)$. Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\Bbb{R}^{2+1}$. I will talk about our proof and explain several important ingredients such as induction on scales and an incidence type theorem.

December 1, 2022, Thursday, 2:00-3:00pm, Hill 525

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Speaker: Rita Teixeira da Costa
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Princeton University

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Title:
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Decay for the Teukolsky equation on subextremal Kerr black holes

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Abstract:
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The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family. We show that solutions arising from suitably regular initial data remain bounded and decay inverse polynomially in time. Our proof holds for the entire subextremal range of Kerr black hole parameters, $|a|\; <$ M. This is joint work with Yakov Shlapentokh-Rothman (Toronto).

November 17, 2022, Thursday, 2:00-3:00pm, Hill 525

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Speaker: Theodore Drivas
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Stony Brook University

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Title:
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Remarks on the long-time dynamics of 2D Euler

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Abstract:
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We will discuss some old and new results concerning the long-time behavior of solutions to the two-dimensional incompressible Euler equations. Specifically, we discuss whether steady states can be isolated, wandering for solutions starting nearby certain steady states, singularity formation at infinite time, and finally some results/conjectures on the infinite-time limit near and far from equilibrium.

November 10, 2022, Thursday, 3:30-4:30pm [SPECIAL TIME AND ROOM], Hill 705 and Zoom (link
here)

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Speaker: Elena Giorgi
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Columbia University

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Title:
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Physical-space estimates on black hole perturbations

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Abstract:
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Most works on the analysis of the wave equation on Kerr black holes rely on a combination of the vector field method and Fourier decomposition, with the notable exception of a generalized vector field method introduced by Andersson-Blue. Their method allows for commutation with second order differential operators entirely in physical-space by supplementing the Killing vector fields with the Carter operator of Kerr to obtain a local energy decay identity at the level of three derivatives of the solution for sufficiently small |a|. In this talk I will describe the main ideas of Andersson-Blue’s method and explain its advantages in two recent applications where physical space-estimates have been crucial: the linear stability of Kerr-Newman black hole to coupled gravitational-electromagnetic perturbations and our proof of the non-linear stability of the slowly rotating Kerr family with Klainerman-Szeftel.

November 3, 2022, Thursday, 2:00-3:00pm, Hill 525

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Speaker: Allen Fang
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Princeton University

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Title:
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A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter

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Abstract:
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The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.

October 20, 2022, Thursday, 2:00-3:00pm, Zoom (online talk).

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Speaker: Pierre Germain
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Imperial College London

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Title:
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The mathematics of wave turbulence, from microscopic systems to turbulent spectra

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Abstract:
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The theory of wave turbulence has been developed by physicists to describe the turbulent behavior of (weakly) nonlinear wave equations. Much like the famous Kolmogorov theory, this theory is able to predict the distribution of energy in turbulent systems; but it is much more precise, in that it gives a kinetic equation governing energy exchanges. From a mathematical viewpoint, this is a very attractive entry point into the mysterious world of turbulence. I will present recent progress on these questions.

October 13, 2022, Thursday, 2:00-3:00pm, Hill 525

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Speaker: Dan Ginsberg
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Princeton University

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Title:
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The stability of model shocks and the Landau law of decay

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Abstract:
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It is well-known that in three space dimensions, smooth solutions to the equations describing a compressible gas can break down in finite time. One type of singularity which can arise is known as a "shock", which is a hypersurface of discontinuity across which the integral forms of conservation of mass and momentum hold and through which there is nonzero mass flux. One can find approximate solutions to the equations of motion which describe expanding spherical shocks. We use these model solutions to construct global-in-time solutions to the irrotational compressible Euler equations with shocks. This is joint work with Igor Rodnianski.

September 13, 2022, Tuesday, 2:00-3:00pm, Hill 705

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Speaker: Gunther Uhlmann
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University of Washington

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Title:
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Seeing Through Space-Time

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Abstract:
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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. The method applies to several other inverse problems for
nonlinear equations, for example the equations of nonlinear
acoustics. No previous knowledge of Lorentzian geometry will be
assumed.