July 31-August 4, 2017 |
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| Mon, Jul 31, 1pm | Martin Reiris (Montevideo): "A classification theorem for static solutions of the vacuum Einstein equations" The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. We extend this result by proving a classification theorem for all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without making any further assumption on the topology or the asymptotic. It is shown that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole, or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Myers/Korotkin-Nicolai black holes. In a broad sense, the theorem classifies all the static vacuum black holes in 3+1-dimensions. |
| Wed, Aug 2, 3:30pm |
Katharina Radermacher (KTH Stockholm): "Strong Cosmic Censorship and the initial singularity in Bianchi spacetimes" For given initial data to Einstein's field equations, one can find a spacetime solving these equations, and one can do so in a unique way (up to isometries) if one assumes the spacetime to be maximal globally hyperbolic. Both statements were proven by Choquet-Bruhat and Geroch in the 1950s and 60s. When dropping the additional condition of global hyperbolicity, it is an open question whether one can extend this spacetime, possibly in a non-unique way. Strong Cosmic Censorship conjectures that no such extension exists, at least not for generic initial data. |
| Fri, Aug 4, 1pm |
Markus Khuri (SUNY): "Existence and Uniqueness for Near-Horizon Geometries in the Cosmological Setting" We prove existence and uniqueness results for all possible mutli-axisymmetic near-horizon geometries with a given horizon topology in the presence of a positive cosmological constant. An explicit classification has previously been given in the Λ=0 case, using different methods. Applications will also be discussed. This is joint work with H. Kunduri and A. Alaee. |
August 7-11, 2017 |
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| Mon, Aug 7, 1pm | Eric Woolgar (Alberta): "Lorentzian Bakry-Émery Theory" In a recent seminar, Khuri observed that Riemannian Bakry-Émery geometry may have a role to play in the study of near-horizon geometries. I will describe Bakry-Émery geometry and show how the Lorentzian version arises in scalar-tensor gravitation. A fairly complete picture of Lorentzian Bakry-Émery geodesic geometry is now available. J Case initiated its study several years ago, giving an extension of the Hawking-Penrose singularity theorem and timelike splitting theorem. Galloway and the speaker then found a version of the Hawking cosmological singularity theorem for Bakry-Émery geometry with infinite synthetic dimension. Wylie and the speaker have now extended these results to all values of the synthetic dimension, including negative values, save for a gap a≤N≤n where a=1 or a=2 depending on the theorem. The techniques are standard applications of geodesic geometry, with a few tricks. |
| Wed, Aug 9, 3:30pm |
Annegret Burtscher (Bonn): "On the asymptotic behavior of static perfect fluids" We are interested in the global behavior of solutions of the static Einstein-Euler equations in spherical symmetry, a system of two singular nonlinear ordinary differential equations which is used for stellar models in astrophysics. Depending on the equation of state, such solutions have either finite or infinite extend. All solutions with finite extend and some very special solutions with infinite extend are geometrically well-understood: they are asymptotically flat. Unfortunately, most solutions do not fall in this category and up to now a geometric framework describing their behavior as the radius tends to infinity is missing. We employ dynamical systems analysis to investigate the asymptotic behavior of solutions to the initial value problem for linear and polytropic equations of state and relate it to the asymptotic behavior of global monopoles. This is joint work with Lars Andersson. |
| Thu, Aug 10, 3:30pm |
Gilbert Weinstein (Ariel University): "Bi-axisymmetric stationary solutions to the vacuum Einstein equation with non-spherical horizons" In the last 15 years, there has been much progress on higher dimensional solutions to the Einstein equation, much of it from the physics community. They are particularly interesting as, unlike 4 dimensional spacetimes, the horizon is no longer restricted to being diffeomorphic to the sphere, as demonstrated by the celebrated black ring solution of Emparan and Reall. Using the Weyl-Papapetrou coordinates and harmonic maps, we show the existence of stationary solutions to the 5 dimensional vacuum Einstein equation, which are bi-axisymmetric solutions with lens space horizons. This is a joint project with Marcus Khuri and Sumio Yamada. |
| Fri, Aug 11, 1pm |
Peter Topping (Warwick): "Ricci flow and Ricci limit spaces" Ricci flow theory has been developing rapidly over the last couple of years, with the ability to handle Ricci flows with unbounded curvature finally becoming a reality. This is vastly expanding the range of potential applications. I will describe some recent work in this direction with Miles Simon. |
August 14-18, 2017 |
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| Mon, Aug 14, 1pm |
Luc Nguyen (Oxford): "Existence and uniqueness of Green's functions to a nonlinear Yamabe problem" The Yamabe problem asks to find in a conformal class of metrics on a compact manifold M a metric of constant scalar curvature. In this context, the Green's function with a given pole p in M corresponds to a complete asymptotically flat metric of zero scalar curvature on M - {p}. We discuss existence and uniqueness of similar objects when the scalar curvature is replaced by other fully nonlinear conformal curvature quantities. |
August 21-25, 2017 |
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| Mon, Aug 21, 3:30pm |
Stefan Haller (Vienna): "The heat equation on filtered manifolds" Elliptic operators have proved to be a powerful tool for relating geometry on manifolds to global topological properties. A large number of geometric structures can be described in terms of filtered manifolds, contact structures arguably being among the best known. Interesting differential operators associated with filtered manifolds tend to be hypoelliptic rather than elliptic. To study these operators, the classical Heisenberg calculus on contact manifolds has recently been generalized to all filtered manifolds. In this talk we will present a Rockland type theorem, characterizing the existence of a parametrix in this calculus. As a first application of this result, we will discuss the heat kernel asymptotics for Rockland differential operators on general filtered manifolds, and address related statements, including the McKean--Singer index formula and Weyl's law. We will also discuss applications to a particular geometry in five dimensions, related to the exceptional Lie group G2, which has been a motivating example in our investigations. This talk is based on joint work with Shantanu Dave. |
| Tue, Aug 22, 3:30pm |
Hubert Bray (Duke): "Flatly Foliated Relativity: A Stepping Stone between Special and General Relativity" What if matter curved spacetime, but not space? That is, suppose the spacetime metric and matter fields minimized the same action, but with the constraint that spacetime is foliated by flat 3-dimensional Euclidean spaces. Interestingly enough, both the Schwarzschild spacetimes and FLRW cosmologies (with k = 0) admit flat foliations, so the Big Bang and black holes still exist in this theory. On the other hand, gravitational waves do not exist in this theory. More generally, the hyperbolic Einstein equation is replaced by elliptic equations, which naturally are still nonlinear. Thus, in this theory, gravity still exists, but is transmitted instantaneously through the rigid flat slices. We present this new theory, which in some sense is 2/3 of the way from special relativity towards general relativity, as an interesting stepping stone for understanding general relativity better. |
| Wed, Aug 23, 3:30pm |
Volker Branding (University of Vienna): "The supersymmetric nonlinear sigma model: Harmonic maps coupled to spinor fields" We will discuss the functional of the supersymmetric nonlinear sigma model as a geometric variational problem. In the case of a Riemannian domain its critical points couple the harmonic map equation to spinor fields, these became known as Dirac-harmonic maps and variants thereof in the mathematics literature. We will give an overview of the current known results on the geometric and analytic properties of the critical points. Finally, we will present recent work on the existence problem for Dirac-wave maps, which are Dirac-harmonic maps from a Lorentzian domain. |
| Thu, Aug 24, 3:30pm |
Juan Valiente-Kroon (Queen Mary): "On the construction of anti-de Sitter-like spacetimes" In this talk I discuss a new approach to the systematic construction of 4-dimensional anti-de Sitter-like spacetimes in the tracefree matter and vacuum cases by means of an initial-boundary value problem for Friedrich's conformal field equations. This construction allows to prescribe as Dirichlet boundary data the 3-metric of the conformal boundary of the spacetime and contains, as a particular case, reflective boundary conditions. The hyperbolic reduction of the conformal equations used in the analysis leads to wave equations for the various conformal fields and makes use generalised harmonic coordinates and is close in spirit to the framework often used in numerical Relativity. Accordingly, it is hoped it will be simpler to implement numerically than other conformal approaches to the construction of anti-de Sitter-like spacetimes. |
| Fri, Aug 25, 1pm |
Jeremie Joudioux (Vienna): "The stability of Minkowski space as a solution to the Einstein-Vlasov system" Joint work with David Fajman (Vienna) and Jacques Smulevici (Orsay) (arXiv:1707.06141). We discuss in this talk the proof of the stability of Minkowski space as a solution to the Einstein-Vlasov system. This proof is based on the construction of appropriate commutators with the transport operator. This method is known as the modified vector field method, and was developed in earlier work by the same authors for the massive Vlasov-Nordström system. It is based on the approach by Klainerman for the wave equation. We work in the wave gauge and a hyperboloidal foliation similar to the work of LeFloch-Ma on the stability of Minkowski spacetime as a solution to the Einstein-Klein-Gordon system. In this talk, we focus on the commutation properties of the transport equation, and discuss in detail the structure of the commutation formula leading to the necessary decay estimates for the Vlasov field and their consequences for the analysis of the Einstein-Vlasov system. |
September 4-8, 2017 |
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| Wed, Sep 6, 3:30pm |
Hakan Andreasson (Chalmers): "Approximating gravitational collapse for dust with Vlasov matter" In the seminal work by Oppenheimer and Snyder from 1939 it is shown that a homogeneous ball of dust undergoes gravitational collapse. I will present a result which shows that this gravitational collapse can be approximated arbitrary well by solutions to the Einstein-Vlasov system. Extensions of this result to the inhomogeneous case will also be discussed. In particular, there exist inhomogeneous data for dust which give rise to naked singularities and it is thus important to understand the relation between the dust solutions and the solutions to the Einstein-Vlasov system in the context of the weak cosmic censorship conjecture. This is a joint work with Gerhard Rein. |
| Fri, Sep 8, 1pm |
Tim Paetz (Vienna): "Analysis of a Bianchi-like equation satisfied by the Mars-Simon tensor" The Mars-Simon tensor (MST) plays an important role to e.g. provide gauge invariant characterizations of the Kerr-NUT-(A)(dS) family, or to establish certain Kerr uniqueness results. It satisfies a Bianchi-like equation. In this talk we analyze this equation in close analogy to the Bianchi equation, in particular one can show that the constraints are preserved supposing that a generalized Buchdahl condition holds. This permits the systematic construction of solutions to this equation in terms of a well-posed Cauchy problem. A particular emphasis lies on the asymptotic Cauchy problem, where data are prescribed on a spacelike Scri (i.e. for positive cosmological constant). In contrast to the Bianchi equation, the MST equation is of Fuchsian type at Scri, for which we establish existence and uniqueness results. This is joint work with Florian Beyer. |